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Theorem mbfimaopnlem 25612

Description: Lemma for mbfimaopn 25613. (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 2736	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		mbfimaopn.1	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	1, 2, 3	cnrehmeo 24907	. . . . . . 7 ⊢ 𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽)
5		hmeocn 23704	. . . . . . 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 23209	. . . . . 6 ⊢ ((𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,))) Cn 𝐽) ∧ 𝐴 ∈ 𝐽) → (^◡𝐺 “ 𝐴) ∈ ((topGen‘ran (,)) ×_t (topGen‘ran (,))))
8	6, 7	mpan 690	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → (^◡𝐺 “ 𝐴) ∈ ((topGen‘ran (,)) ×_t (topGen‘ran (,))))
9		mbfimaopn.3	. . . . . . . . 9 ⊢ 𝐵 = ((,) “ (ℚ × ℚ))
10	9	fveq2i 6837	. . . . . . . 8 ⊢ (topGen‘𝐵) = (topGen‘((,) “ (ℚ × ℚ)))
11	10	tgqioo 24744	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘𝐵)
12	11, 11	oveq12i 7370	. . . . . 6 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = ((topGen‘𝐵) ×_t (topGen‘𝐵))
13		qtopbas 24703	. . . . . . . 8 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
14	9, 13	eqeltri 2832	. . . . . . 7 ⊢ 𝐵 ∈ TopBases
15		txbasval 23550	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → ((topGen‘𝐵) ×_t (topGen‘𝐵)) = (𝐵 ×_t 𝐵))
16	14, 14, 15	mp2an 692	. . . . . 6 ⊢ ((topGen‘𝐵) ×_t (topGen‘𝐵)) = (𝐵 ×_t 𝐵)
17		mbfimaopn.4	. . . . . . . 8 ⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
18	17	txval 23508	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → (𝐵 ×_t 𝐵) = (topGen‘𝐾))
19	14, 14, 18	mp2an 692	. . . . . 6 ⊢ (𝐵 ×_t 𝐵) = (topGen‘𝐾)
20	12, 16, 19	3eqtri 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = (topGen‘𝐾)
21	8, 20	eleqtrdi 2846	. . . 4 ⊢ (𝐴 ∈ 𝐽 → (^◡𝐺 “ 𝐴) ∈ (topGen‘𝐾))
22	17	txbas 23511	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → 𝐾 ∈ TopBases)
23	14, 14, 22	mp2an 692	. . . . 5 ⊢ 𝐾 ∈ TopBases
24		eltg3 22906	. . . . 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 218	. . 3 ⊢ (𝐴 ∈ 𝐽 → ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡))
27	26	adantl 481	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → ∃𝑡(𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡))
28	1	cnref1o 12898	. . . . . . . 8 ⊢ 𝐺:(ℝ × ℝ)–1-1-onto→ℂ
29		f1ofo 6781	. . . . . . . 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 4894	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
32	3	cnfldtopon 24726	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
33	32	toponunii 22860	. . . . . . . . 9 ⊢ ℂ = ∪ 𝐽
34	31, 33	sseqtrrdi 3975	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ℂ)
35	34	ad2antlr 727	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 ⊆ ℂ)
36		foimacnv 6791	. . . . . . 7 ⊢ ((𝐺:(ℝ × ℝ)–onto→ℂ ∧ 𝐴 ⊆ ℂ) → (𝐺 “ (^◡𝐺 “ 𝐴)) = 𝐴)
37	30, 35, 36	sylancr 587	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = 𝐴)
38		simprr 772	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐺 “ 𝐴) = ∪ 𝑡)
39	38	imaeq2d 6019	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = (𝐺 “ ∪ 𝑡))
40		imauni 7192	. . . . . . 7 ⊢ (𝐺 “ ∪ 𝑡) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)
41	39, 40	eqtrdi 2787	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
42	37, 41	eqtr3d 2773	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
43	42	imaeq2d 6019	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)))
44		imaiun 7191	. . . 4 ⊢ (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤))
45	43, 44	eqtrdi 2787	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)))
46		ssdomg 8937	. . . . . . 7 ⊢ (𝐾 ∈ TopBases → (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾))
47	23, 46	ax-mp 5	. . . . . 6 ⊢ (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾)
48		omelon 9555	. . . . . . . . . . 11 ⊢ ω ∈ On
49		nnenom 13903	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
50	49	ensymi 8941	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
51		isnumi 9858	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
52	48, 50, 51	mp2an 692	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
53		qnnen 16138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ≈ ℕ
54		xpen 9068	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ))
55	53, 53, 54	mp2an 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ)
56		xpnnen 16136	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ × ℕ) ≈ ℕ
57	55, 56	entri 8945	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ≈ ℕ
58	57, 49	entr2i 8946	. . . . . . . . . . . . . . . . 17 ⊢ ω ≈ (ℚ × ℚ)
59		isnumi 9858	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card)
60	48, 58, 59	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (ℚ × ℚ) ∈ dom card
61		ioof 13363	. . . . . . . . . . . . . . . . . 18 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
62		ffun 6665	. . . . . . . . . . . . . . . . . 18 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
63	61, 62	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun (,)
64		qssre 12872	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ⊆ ℝ
65		ressxr 11176	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℝ^*
66	64, 65	sstri 3943	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ ⊆ ℝ^*
67		xpss12 5639	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℚ ⊆ ℝ^* ∧ ℚ ⊆ ℝ^) → (ℚ × ℚ) ⊆ (ℝ^ × ℝ^*))
68	66, 66, 67	mp2an 692	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ⊆ (ℝ^* × ℝ^*)
69	61	fdmi 6673	. . . . . . . . . . . . . . . . . 18 ⊢ dom (,) = (ℝ^* × ℝ^*)
70	68, 69	sseqtrri 3983	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ × ℚ) ⊆ dom (,)
71		fores 6756	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun (,) ∧ (ℚ × ℚ) ⊆ dom (,)) → ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)))
72	63, 70, 71	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ))
73		fodomnum 9967	. . . . . . . . . . . . . . . 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 5119	. . . . . . . . . . . . . 14 ⊢ 𝐵 ≼ (ℚ × ℚ)
76		domentr 8950	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ℕ) → 𝐵 ≼ ℕ)
77	75, 57, 76	mp2an 692	. . . . . . . . . . . . 13 ⊢ 𝐵 ≼ ℕ
78	14	elexi 3463	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
79	78	xpdom1 9004	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (𝐵 × 𝐵) ≼ (ℕ × 𝐵))
80	77, 79	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐵 × 𝐵) ≼ (ℕ × 𝐵)
81		nnex 12151	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
82	81	xpdom2 9000	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (ℕ × 𝐵) ≼ (ℕ × ℕ))
83	77, 82	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℕ × 𝐵) ≼ (ℕ × ℕ)
84		domtr 8944	. . . . . . . . . . . 12 ⊢ (((𝐵 × 𝐵) ≼ (ℕ × 𝐵) ∧ (ℕ × 𝐵) ≼ (ℕ × ℕ)) → (𝐵 × 𝐵) ≼ (ℕ × ℕ))
85	80, 83, 84	mp2an 692	. . . . . . . . . . 11 ⊢ (𝐵 × 𝐵) ≼ (ℕ × ℕ)
86		domentr 8950	. . . . . . . . . . 11 ⊢ (((𝐵 × 𝐵) ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → (𝐵 × 𝐵) ≼ ℕ)
87	85, 56, 86	mp2an 692	. . . . . . . . . 10 ⊢ (𝐵 × 𝐵) ≼ ℕ
88		numdom 9948	. . . . . . . . . 10 ⊢ ((ℕ ∈ dom card ∧ (𝐵 × 𝐵) ≼ ℕ) → (𝐵 × 𝐵) ∈ dom card)
89	52, 87, 88	mp2an 692	. . . . . . . . 9 ⊢ (𝐵 × 𝐵) ∈ dom card
90		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
91		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
92		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
93	91, 92	xpex 7698	. . . . . . . . . . 11 ⊢ (𝑥 × 𝑦) ∈ V
94	90, 93	fnmpoi 8014	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) Fn (𝐵 × 𝐵)
95		dffn4 6752	. . . . . . . . . 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 230	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)):(𝐵 × 𝐵)–onto→ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
97		fodomnum 9967	. . . . . . . . 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 8944	. . . . . . . 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 692	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ≼ ℕ
101	17, 100	eqbrtri 5119	. . . . . 6 ⊢ 𝐾 ≼ ℕ
102		domtr 8944	. . . . . 6 ⊢ ((𝑡 ≼ 𝐾 ∧ 𝐾 ≼ ℕ) → 𝑡 ≼ ℕ)
103	47, 101, 102	sylancl 586	. . . . 5 ⊢ (𝑡 ⊆ 𝐾 → 𝑡 ≼ ℕ)
104	103	ad2antrl 728	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝑡 ≼ ℕ)
105	17	eleq2i 2828	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐾 ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)))
106	90, 93	elrnmpo 7494	. . . . . . . . 9 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦))
107	105, 106	bitri 275	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐾 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦))
108		elin 3917	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)))
109		mbff 25582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
110	109	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹:dom 𝐹⟶ℂ)
111		fvco3 6933	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → ((ℜ ∘ 𝐹)‘𝑧) = (ℜ‘(𝐹‘𝑧)))
112	110, 111	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((ℜ ∘ 𝐹)‘𝑧) = (ℜ‘(𝐹‘𝑧)))
113	112	eleq1d 2821	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ↔ (ℜ‘(𝐹‘𝑧)) ∈ 𝑥))
114		fvco3 6933	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → ((ℑ ∘ 𝐹)‘𝑧) = (ℑ‘(𝐹‘𝑧)))
115	110, 114	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → ((ℑ ∘ 𝐹)‘𝑧) = (ℑ‘(𝐹‘𝑧)))
116	115	eleq1d 2821	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦 ↔ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦))
117	113, 116	anbi12d 632	. . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
119		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℜ‘𝑤) = (ℜ‘(𝐹‘𝑧)))
120		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℑ‘𝑤) = (ℑ‘(𝐹‘𝑧)))
121	119, 120	opeq12d 4837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = (𝐹‘𝑧) → ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩ = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
122	1	cnrecnv 15088	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ^◡𝐺 = (𝑤 ∈ ℂ ↦ ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩)
123		opex 5412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ V
124	121, 122, 123	fvmpt 6941	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑧) ∈ ℂ → (^◡𝐺‘(𝐹‘𝑧)) = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
125	118, 124	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (^◡𝐺‘(𝐹‘𝑧)) = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
126	125	eleq1d 2821	. . . . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . 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 5660	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ (𝑥 × 𝑦) ↔ ((ℜ‘(𝐹‘𝑧)) ∈ 𝑥 ∧ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦))
130		f1ocnv 6786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ))
131		f1ofn 6775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → ^◡𝐺 Fn ℂ)
132	28, 130, 131	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ ^◡𝐺 Fn ℂ
133		elpreima 7003	. . . . . . . . . . . . . . . . . . 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 6164	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡^◡𝐺 “ (𝑥 × 𝑦)) = (𝐺 “ (𝑥 × 𝑦))
136	135	eleq2i 2828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑧) ∈ (^◡^◡𝐺 “ (𝑥 × 𝑦)) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))
137	134, 136	bitr3i 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑧) ∈ ℂ ∧ (^◡𝐺‘(𝐹‘𝑧)) ∈ (𝑥 × 𝑦)) ↔ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))
138	128, 129, 137	3bitr3g 313	. . . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . 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 15035	. . . . . . . . . . . . . . . . . . 19 ⊢ ℜ:ℂ⟶ℝ
142		fco 6686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
143	141, 109, 142	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹):dom 𝐹⟶ℝ)
144		ffn 6662	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → (ℜ ∘ 𝐹) Fn dom 𝐹)
145		elpreima 7003	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℜ ∘ 𝐹) Fn dom 𝐹 → (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥)))
146	143, 144, 145	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥)))
147		imf 15036	. . . . . . . . . . . . . . . . . . 19 ⊢ ℑ:ℂ⟶ℝ
148		fco 6686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:dom 𝐹⟶ℂ) → (ℑ ∘ 𝐹):dom 𝐹⟶ℝ)
149	147, 109, 148	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ MblFn → (ℑ ∘ 𝐹):dom 𝐹⟶ℝ)
150		ffn 6662	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ ∘ 𝐹):dom 𝐹⟶ℝ → (ℑ ∘ 𝐹) Fn dom 𝐹)
151		elpreima 7003	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ ∘ 𝐹) Fn dom 𝐹 → (𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)))
152	149, 150, 151	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦) ↔ (𝑧 ∈ dom 𝐹 ∧ ((ℑ ∘ 𝐹)‘𝑧) ∈ 𝑦)))
153	146, 152	anbi12d 632	. . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 6662	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
158		elpreima 7003	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn dom 𝐹 → (𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
159	109, 157, 158	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ↔ (𝑧 ∈ dom 𝐹 ∧ (𝐹‘𝑧) ∈ (𝐺 “ (𝑥 × 𝑦)))))
160	159	adantr 480	. . . . . . . . . . . . . 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 311	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ 𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦)))))
162	108, 161	bitrid 283	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑧 ∈ ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ 𝑧 ∈ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦)))))
163	162	eqrdv 2734	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
164		ismbfcn 25586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:dom 𝐹⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
165	109, 164	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ MblFn → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))
166	165	ibi 267	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ MblFn → ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))
167	166	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (ℜ ∘ 𝐹) ∈ MblFn)
168		ismbf 25585	. . . . . . . . . . . . . . . 16 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
169	143, 168	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
170	167, 169	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ MblFn → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
171	170	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
172		imassrn 6030	. . . . . . . . . . . . . . 15 ⊢ ((,) “ (ℚ × ℚ)) ⊆ ran (,)
173	9, 172	eqsstri 3980	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ ran (,)
174		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
175	173, 174	sselid 3931	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ran (,))
176		rsp 3224	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ran (,)(^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol → (𝑥 ∈ ran (,) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol))
177	171, 175, 176	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)
178	166	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → (ℑ ∘ 𝐹) ∈ MblFn)
179		ismbf 25585	. . . . . . . . . . . . . . . 16 ⊢ ((ℑ ∘ 𝐹):dom 𝐹⟶ℝ → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
180	149, 179	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ MblFn → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
181	178, 180	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ MblFn → ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
182	181	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
183		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
184	173, 183	sselid 3931	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ran (,))
185		rsp 3224	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ ran (,)(^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol → (𝑦 ∈ ran (,) → (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol))
186	182, 184, 185	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol)
187		inmbl 25499	. . . . . . . . . . . 12 ⊢ (((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑦) ∈ dom vol) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ∈ dom vol)
188	177, 186, 187	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ∈ dom vol)
189	163, 188	eqeltrrd 2837	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ∈ dom vol)
190		imaeq2 6015	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥 × 𝑦) → (𝐺 “ 𝑤) = (𝐺 “ (𝑥 × 𝑦)))
191	190	imaeq2d 6019	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
192	191	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 × 𝑦) → ((^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol ↔ (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ∈ dom vol))
193	189, 192	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
194	193	rexlimdvva 3193	. . . . . . . 8 ⊢ (𝐹 ∈ MblFn → (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
195	107, 194	biimtrid 242	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → (𝑤 ∈ 𝐾 → (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
196	195	ralrimiv 3127	. . . . . 6 ⊢ (𝐹 ∈ MblFn → ∀𝑤 ∈ 𝐾 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
197		ssralv 4002	. . . . . 6 ⊢ (𝑡 ⊆ 𝐾 → (∀𝑤 ∈ 𝐾 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol))
198	196, 197	mpan9 506	. . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝑡 ⊆ 𝐾) → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
199	198	ad2ant2r 747	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
200		iunmbl2 25514	. . . 4 ⊢ ((𝑡 ≼ ℕ ∧ ∀𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol) → ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
201	104, 199, 200	syl2anc 584	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
202	45, 201	eqeltrd 2836	. 2 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) ∈ dom vol)
203	27, 202	exlimddv 1936	1 ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 ∩ cin 3900 ⊆ wss 3901 𝒫 cpw 4554 ⟨cop 4586 ∪ cuni 4863 ∪ ciun 4946 class class class wbr 5098 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 ∘ ccom 5628 Oncon0 6317 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ωcom 7808 ≈ cen 8880 ≼ cdom 8881 cardccrd 9847 ℂcc 11024 ℝcr 11025 ici 11028 + caddc 11029 · cmul 11031 ℝ^*cxr 11165 ℕcn 12145 ℚcq 12861 (,)cioo 13261 ℜcre 15020 ℑcim 15021 TopOpenctopn 17341 topGenctg 17357 ℂ_fldccnfld 21309 TopBasesctb 22889 Cn ccn 23168 ×_t ctx 23504 Homeochmeo 23697 volcvol 25420 MblFncmbf 25571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cn 23171 df-cnp 23172 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25421 df-vol 25422 df-mbf 25576

This theorem is referenced by: mbfimaopn 25613

W3C validator