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Theorem mbfimaopnlem 25865

Description: Lemma for mbfimaopn 25866. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
mbfimaopn.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)
mbfimaopn.2	⊢ 𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
mbfimaopn.3	⊢ 𝐵 = ((,) “ (ℚ × ℚ))
mbfimaopn.4	⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))

**Assertion**
Ref	Expression
mbfimaopnlem	⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ dom vol)

Distinct variable groups: 𝑥,𝐴 𝑥,𝑦,𝐵 𝑥,𝐹,𝑦 𝑥,𝐺,𝑦 𝑥,𝐽,𝑦 Allowed substitution hints: 𝐴(𝑦) 𝐾(𝑥, 𝑦)

Proof of Theorem mbfimaopnlem Dummy variables 𝑡 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfimaopn.2	. . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2		eqid 2765	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		mbfimaopn.1	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	1, 2, 3	cnrehmeo 25163	. . . . . . 7 ⊢ 𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽)
5		hmeocn 23968	. . . . . . 7 ⊢ (𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽) → 𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ 𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽)
7		cnima 23472	. . . . . 6 ⊢ ((𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽) ∧ 𝐴 ∈ 𝐽) → (^◡𝐺 “ 𝐴) ∈ ((topGen‘ran (,)) ×_t (topGen‘ran (,))))
8	6, 7	mpan 703	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → (^◡𝐺 “ 𝐴) ∈ ((topGen‘ran (,)) ×_t (topGen‘ran (,))))
9		mbfimaopn.3	. . . . . . . . 9 ⊢ 𝐵 = ((,) “ (ℚ × ℚ))
10	9	fveq2i 6888	. . . . . . . 8 ⊢ (topGen‘𝐵) = (topGen‘((,) “ (ℚ × ℚ)))
11	10	tgqioo 25008	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘𝐵)
12	11, 11	oveq12i 7431	. . . . . 6 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = ((topGen‘𝐵) ×_t (topGen‘𝐵))
13		qtopbas 24967	. . . . . . . 8 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
14	9, 13	eqeltri 2861	. . . . . . 7 ⊢ 𝐵 ∈ TopBases
15		txbasval 23814	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → ((topGen‘𝐵) ×_t (topGen‘𝐵)) = (𝐵 ×_t 𝐵))
16	14, 14, 15	mp2an 705	. . . . . 6 ⊢ ((topGen‘𝐵) ×_t (topGen‘𝐵)) = (𝐵 ×_t 𝐵)
17		mbfimaopn.4	. . . . . . . 8 ⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
18	17	txval 23772	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → (𝐵 ×_t 𝐵) = (topGen‘𝐾))
19	14, 14, 18	mp2an 705	. . . . . 6 ⊢ (𝐵 ×_t 𝐵) = (topGen‘𝐾)
20	12, 16, 19	3eqtri 2792	. . . . 5 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = (topGen‘𝐾)
21	8, 20	eleqtrdi 2875	. . . 4 ⊢ (𝐴 ∈ 𝐽 → (^◡𝐺 “ 𝐴) ∈ (topGen‘𝐾))
22	17	txbas 23775	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → 𝐾 ∈ TopBases)
23	14, 14, 22	mp2an 705	. . . . 5 ⊢ 𝐾 ∈ TopBases
24		eltg3 23169	. . . . 5 ⊢ (𝐾 ∈ TopBases → ((^◡𝐺 “ 𝐴) ∈ (topGen‘𝐾) ↔ ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)))
25	23, 24	ax-mp 5	. . . 4 ⊢ ((^◡𝐺 “ 𝐴) ∈ (topGen‘𝐾) ↔ ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡))
26	21, 25	sylib 221	. . 3 ⊢ (𝐴 ∈ 𝐽 → ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡))
27	26	adantl 487	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡))
28	1	cnref1o 13025	. . . . . . . 8 ⊢ 𝐺:(ℝ × ℝ)–1-1-onto→ℂ
29		f1ofo 6832	. . . . . . . 8 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)–onto→ℂ)
30	28, 29	ax-mp 5	. . . . . . 7 ⊢ 𝐺:(ℝ × ℝ)–onto→ℂ
31		elssuni 4906	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
32	3	cnfldtopon 24990	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
33	32	toponunii 23123	. . . . . . . . 9 ⊢ ℂ = ∪ 𝐽
34	31, 33	sseqtrrdi 3979	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ℂ)
35	34	ad2antlr 740	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 ⊆ ℂ)
36		foimacnv 6842	. . . . . . 7 ⊢ ((𝐺:(ℝ × ℝ)–onto→ℂ ∧ 𝐴 ⊆ ℂ) → (𝐺 “ (^◡𝐺 “ 𝐴)) = 𝐴)
37	30, 35, 36	sylancr 599	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = 𝐴)
38		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐺 “ 𝐴) = ∪ 𝑡)
39	38	imaeq2d 6064	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = (𝐺 “ ∪ 𝑡))
40		imauni 7249	. . . . . . 7 ⊢ (𝐺 “ ∪ 𝑡) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)
41	39, 40	eqtrdi 2816	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
42	37, 41	eqtr3d 2802	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
43	42	imaeq2d 6064	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)))
44		imaiun 7248	. . . 4 ⊢ (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤))
45	43, 44	eqtrdi 2816	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)))
46		ssdomg 9003	. . . . . . 7 ⊢ (𝐾 ∈ TopBases → (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾))
47	23, 46	ax-mp 5	. . . . . 6 ⊢ (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾)
48		omelon 9622	. . . . . . . . . . 11 ⊢ ω ∈ On
49		nnenom 14034	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
50	49	ensymi 9007	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
51		isnumi 9948	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
52	48, 50, 51	mp2an 705	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
53		qnnen 16291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ≈ ℕ
54		xpen 9135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ))
55	53, 53, 54	mp2an 705	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ)
56		xpnnen 16289	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ × ℕ) ≈ ℕ
57	55, 56	entri 9011	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ≈ ℕ
58	57, 49	entr2i 9012	. . . . . . . . . . . . . . . . 17 ⊢ ω ≈ (ℚ × ℚ)
59		isnumi 9948	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card)
60	48, 58, 59	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ (ℚ × ℚ) ∈ dom card
61		ioof 13490	. . . . . . . . . . . . . . . . . 18 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
62		ffun 6712	. . . . . . . . . . . . . . . . . 18 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
63	61, 62	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun (,)
64		qssre 12999	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ⊆ ℝ
65		ressxr 11268	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℝ^*
66	64, 65	sstri 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ ⊆ ℝ^*
67		xpss12 5678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℚ ⊆ ℝ^* ∧ ℚ ⊆ ℝ^) → (ℚ × ℚ) ⊆ (ℝ^ × ℝ^*))
68	66, 66, 67	mp2an 705	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ⊆ (ℝ^* × ℝ^*)
69	61	fdmi 6721	. . . . . . . . . . . . . . . . . 18 ⊢ dom (,) = (ℝ^* × ℝ^*)
70	68, 69	sseqtrri 3987	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ × ℚ) ⊆ dom (,)
71		fores 6806	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun (,) ∧ (ℚ × ℚ) ⊆ dom (,)) → ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)))
72	63, 70, 71	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ))
73		fodomnum 10057	. . . . . . . . . . . . . . . 16 ⊢ ((ℚ × ℚ) ∈ dom card → (((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ)))
74	60, 72, 73	mp2 9	. . . . . . . . . . . . . . 15 ⊢ ((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ)
75	9, 74	eqbrtri 5134	. . . . . . . . . . . . . 14 ⊢ 𝐵 ≼ (ℚ × ℚ)
76		domentr 9016	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ℕ) → 𝐵 ≼ ℕ)
77	75, 57, 76	mp2an 705	. . . . . . . . . . . . 13 ⊢ 𝐵 ≼ ℕ
78	14	elexi 3479	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
79	78	xpdom1 9071	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (𝐵 × 𝐵) ≼ (ℕ × 𝐵))
80	77, 79	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐵 × 𝐵) ≼ (ℕ × 𝐵)
81		nnex 12254	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
82	81	xpdom2 9067	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (ℕ × 𝐵) ≼ (ℕ × ℕ))
83	77, 82	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℕ × 𝐵) ≼ (ℕ × ℕ)
84		domtr 9010	. . . . . . . . . . . 12 ⊢ (((𝐵 × 𝐵) ≼ (ℕ × 𝐵) ∧ (ℕ × 𝐵) ≼ (ℕ × ℕ)) → (𝐵 × 𝐵) ≼ (ℕ × ℕ))
85	80, 83, 84	mp2an 705	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ≼ (ℕ × ℕ)
86		domentr 9016	. . . . . . . . . . 11 ⊢ (((𝐵 × 𝐵) ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → (𝐵 × 𝐵) ≼ ℕ)
87	85, 56, 86	mp2an 705	. . . . . . . . . 10 ⊢ (𝐵 × 𝐵) ≼ ℕ
88		numdom 10038	. . . . . . . . . 10 ⊢ ((ℕ ∈ dom card ∧ (𝐵 × 𝐵) ≼ ℕ) → (𝐵 × 𝐵) ∈ dom card)
89	52, 87, 88	mp2an 705	. . . . . . . . 9 ⊢ (𝐵 × 𝐵) ∈ dom card
90		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
91		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
92		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
93	91, 92	xpex 7758	. . . . . . . . . . 11 ⊢ (𝑥 × 𝑦) ∈ V
94	90, 93	fnmpoi 8073	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) Fn (𝐵 × 𝐵)
95		dffn4 6802	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)):(𝐵 × 𝐵)–onto→ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)))
96	94, 95	mpbi 233	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)):(𝐵 × 𝐵)–onto→ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
97		fodomnum 10057	. . . . . . . . 9 ⊢ ((𝐵 × 𝐵) ∈ dom card → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)):(𝐵 × 𝐵)–onto→ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ (𝐵 × 𝐵)))
98	89, 96, 97	mp2 9	. . . . . . . 8 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ (𝐵 × 𝐵)
99		domtr 9010	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ≼ ℕ) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ ℕ)
100	98, 87, 99	mp2an 705	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ ℕ
101	17, 100	eqbrtri 5134	. . . . . 6 ⊢ 𝐾 ≼ ℕ
102		domtr 9010	. . . . . 6 ⊢ ((𝑡 ≼ 𝐾 ∧ 𝐾 ≼ ℕ) → 𝑡 ≼ ℕ)
103	47, 101, 102	sylancl 598	. . . . 5 ⊢ (𝑡 ⊆ 𝐾 → 𝑡 ≼ ℕ)
104	103	ad2antrl 741	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝑡 ≼ ℕ)
105	17	eleq2i 2857	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐾 ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)))
106	90, 93	elrnmpo 7555	. . . . . . . . 9 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦))
107	105, 106	bitri 278	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐾 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦))
108		elin 3922	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)))
109		mbff 25835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
110	109	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹:dom 𝐹⟶ℂ)
111		fvco3 6985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → ((ℜ ∘ 𝐹)‘𝑧) = (ℜ‘(𝐹‘𝑧)))
112	110, 111	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((ℜ ∘ 𝐹)‘𝑧) = (ℜ‘(𝐹‘𝑧)))
113	112	eleq1d 2850	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ↔ (ℜ‘(𝐹‘𝑧)) ∈ 𝑥))
114		fvco3 6985	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → ((ℑ ∘ 𝐹)‘𝑧) = (ℑ‘(𝐹‘𝑧)))
115	110, 114	sylan 592	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((ℑ ∘ 𝐹)‘𝑧) = (ℑ‘(𝐹‘𝑧)))
116	115	eleq1d 2850	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦 ↔ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦))
117	113, 116	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦) ↔ ((ℜ‘(𝐹‘𝑧)) ∈ 𝑥 ∧ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦)))
118	110	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
119		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℜ‘𝑤) = (ℜ‘(𝐹‘𝑧)))
120		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℑ‘𝑤) = (ℑ‘(𝐹‘𝑧)))
121	119, 120	opeq12d 4848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = (𝐹‘𝑧) → ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩ = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
122	1	cnrecnv 15240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ^◡𝐺 = (𝑤 ∈ ℂ ↦ ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩)
123		opex 5447	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ V
124	121, 122, 123	fvmpt 6993	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧) ∈ ℂ → (^◡𝐺‘(𝐹‘𝑧)) = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
125	118, 124	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (^◡𝐺‘(𝐹‘𝑧)) = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
126	125	eleq1d 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦) ↔ ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ (𝑥 × 𝑦)))
127	118	biantrurd 542	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦) ↔ ((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦))))
128	126, 127	bitr3d 284	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ (𝑥 × 𝑦) ↔ ((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦))))
129		opelxp 5699	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ (𝑥 × 𝑦) ↔ ((ℜ‘(𝐹‘𝑧)) ∈ 𝑥 ∧ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦))
130		f1ocnv 6837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ))
131		f1ofn 6825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → ^◡𝐺 Fn ℂ)
132	28, 130, 131	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ ^◡𝐺 Fn ℂ
133		elpreima 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝐺 Fn ℂ → ((𝐹‘𝑧) ∈ (^◡^◡𝐺 “ (𝑥 × 𝑦)) ↔ ((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦))))
134	132, 133	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑧) ∈ (^◡^◡𝐺 “ (𝑥 × 𝑦)) ↔ ((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦)))
135		imacnvcnv 6209	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡^◡𝐺 “ (𝑥 × 𝑦)) = (𝐺 “ (𝑥 × 𝑦))
136	135	eleq2i 2857	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑧) ∈ (^◡^◡𝐺 “ (𝑥 × 𝑦)) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))
137	134, 136	bitr3i 280	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦)) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))
138	128, 129, 137	3bitr3g 316	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℜ‘(𝐹‘𝑧)) ∈ 𝑥 ∧ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦))))
139	117, 138	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦))))
140	139	pm5.32da 590	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑧 ∈ dom 𝐹 ∧ (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
141		ref 15187	. . . . . . . . . . . . . . . . . . 19 ⊢ ℜ:ℂ⟶ℝ
142		fco 6734	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
143	141, 109, 142	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
144		ffn 6709	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → (ℜ ∘ 𝐹) Fn dom 𝐹)
145		elpreima 7057	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℜ ∘ 𝐹) Fn dom 𝐹 → (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥)))
146	143, 144, 145	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥)))
147		imf 15188	. . . . . . . . . . . . . . . . . . 19 ⊢ ℑ:ℂ⟶ℝ
148		fco 6734	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℑ ∘ 𝐹):dom 𝐹⟶ℝ)
149	147, 109, 148	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ MblFn → (ℑ ∘ 𝐹):dom 𝐹⟶ℝ)
150		ffn 6709	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ ∘ 𝐹):dom 𝐹⟶ℝ → (ℑ ∘ 𝐹) Fn dom 𝐹)
151		elpreima 7057	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ ∘ 𝐹) Fn dom 𝐹 → (𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)))
152	149, 150, 151	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)))
153	146, 152	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ MblFn → ((𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ ((𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥) ∧ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦))))
154		anandi 689	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ dom 𝐹 ∧ (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)) ↔ ((𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥) ∧ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)))
155	153, 154	bitr4di 292	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → ((𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ (𝑧 ∈ dom 𝐹 ∧ (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦))))
156	155	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ (𝑧 ∈ dom 𝐹 ∧ (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦))))
157		ffn 6709	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
158		elpreima 7057	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn dom 𝐹 → (𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
159	109, 157, 158	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
160	159	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
161	140, 156, 160	3bitr4d 314	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ 𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦)))))
162	108, 161	bitrid 286	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑧 ∈ ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ 𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦)))))
163	162	eqrdv 2763	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
164		ismbfcn 25839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:dom 𝐹⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
165	109, 164	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
166	165	ibi 270	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ MblFn → ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))
167	166	simpld 500	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹) ∈ MblFn)
168		ismbf 25838	. . . . . . . . . . . . . . . 16 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
169	143, 168	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
170	167, 169	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ MblFn → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
171	170	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
172		imassrn 6075	. . . . . . . . . . . . . . 15 ⊢ ((,) “ (ℚ × ℚ)) ⊆ ran (,)
173	9, 172	eqsstri 3984	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ ran (,)
174		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
175	173, 174	sselid 3936	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ran (,))
176		rsp 3255	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol → (𝑥 ∈ ran (,) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
177	171, 175, 176	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
178	166	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (ℑ ∘ 𝐹) ∈ MblFn)
179		ismbf 25838	. . . . . . . . . . . . . . . 16 ⊢ ((ℑ ∘ 𝐹):dom 𝐹⟶ℝ → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
180	149, 179	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
181	178, 180	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ MblFn → ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
182	181	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
183		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
184	173, 183	sselid 3936	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ran (,))
185		rsp 3255	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol → (𝑦 ∈ ran (,) → (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
186	182, 184, 185	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
187		inmbl 25752	. . . . . . . . . . . 12 ⊢ (((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ∈ dom vol)
188	177, 186, 187	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ∈ dom vol)
189	163, 188	eqeltrrd 2866	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ∈ dom vol)
190		imaeq2 6060	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥 × 𝑦) → (𝐺 “ 𝑤) = (𝐺 “ (𝑥 × 𝑦)))
191	190	imaeq2d 6064	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
192	191	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 × 𝑦) → ((^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol ↔ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ∈ dom vol))
193	189, 192	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
194	193	rexlimdvva 3224	. . . . . . . 8 ⊢ (𝐹 ∈ MblFn → (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
195	107, 194	biimtrid 245	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → (𝑤 ∈ 𝐾 → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
196	195	ralrimiv 3158	. . . . . 6 ⊢ (𝐹 ∈ MblFn → ∀𝑤 ∈ 𝐾 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
197		ssralv 4007	. . . . . 6 ⊢ (𝑡 ⊆ 𝐾 → (∀𝑤 ∈ 𝐾 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
198	196, 197	mpan9 516	. . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝑡 ⊆ 𝐾) → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
199	198	ad2ant2r 760	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
200		iunmbl2 25767	. . . 4 ⊢ ((𝑡 ≼ ℕ ∧ ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol) → ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
201	104, 199, 200	syl2anc 596	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
202	45, 201	eqeltrd 2865	. 2 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) ∈ dom vol)
203	27, 202	exlimddv 1968	1 ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ dom vol)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4564 ⟨cop 4597 ∪ cuni 4874 ∪ ciun 4958 class class class wbr 5111 × cxp 5661 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 ↾ cres 5665 “ cima 5666 ∘ ccom 5667 Oncon0 6364 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 ωcom 7868 ≈ cen 8946 ≼ cdom 8947 cardccrd 9937 ℂcc 11113 ℝcr 11114 ici 11117 + caddc 11118 · cmul 11120 ℝ^*cxr 11257 ℕcn 12248 ℚcq 12988 (,)cioo 13388 ℜcre 15172 ℑcim 15173 TopOpenctopn 17496 topGenctg 17512 ℂ_fldccnfld 21572 TopBasesctb 23152 Cn ccn 23431 ×_t ctx 23768 Homeochmeo 23961 volcvol 25673 MblFncmbf 25824

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cc 10434 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-acn 9944 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-rlim 15564 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cn 23434 df-cnp 23435 df-tx 23770 df-hmeo 23963 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-ovol 25674 df-vol 25675 df-mbf 25829

This theorem is used by: mbfimaopn 25866

W3C validator