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Theorem mbfimaopnlem 25556

Description: Lemma for mbfimaopn 25557. (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 2729	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		mbfimaopn.1	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	1, 2, 3	cnrehmeo 24851	. . . . . . 7 ⊢ 𝐺 ∈ (((topGen‘ran (,)) ×_t (topGen‘ran (,)))Homeo𝐽)
5		hmeocn 23647	. . . . . . 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 23152	. . . . . 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 6861	. . . . . . . 8 ⊢ (topGen‘𝐵) = (topGen‘((,) “ (ℚ × ℚ)))
11	10	tgqioo 24688	. . . . . . 7 ⊢ (topGen‘ran (,)) = (topGen‘𝐵)
12	11, 11	oveq12i 7399	. . . . . 6 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = ((topGen‘𝐵) ×_t (topGen‘𝐵))
13		qtopbas 24647	. . . . . . . 8 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
14	9, 13	eqeltri 2824	. . . . . . 7 ⊢ 𝐵 ∈ TopBases
15		txbasval 23493	. . . . . . 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 23451	. . . . . . 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 2756	. . . . 5 ⊢ ((topGen‘ran (,)) ×_t (topGen‘ran (,))) = (topGen‘𝐾)
21	8, 20	eleqtrdi 2838	. . . 4 ⊢ (𝐴 ∈ 𝐽 → (^◡𝐺 “ 𝐴) ∈ (topGen‘𝐾))
22	17	txbas 23454	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝐵 ∈ TopBases) → 𝐾 ∈ TopBases)
23	14, 14, 22	mp2an 692	. . . . 5 ⊢ 𝐾 ∈ TopBases
24		eltg3 22849	. . . . 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 12944	. . . . . . . 8 ⊢ 𝐺:(ℝ × ℝ)–1-1-onto→ℂ
29		f1ofo 6807	. . . . . . . 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 4901	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
32	3	cnfldtopon 24670	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
33	32	toponunii 22803	. . . . . . . . 9 ⊢ ℂ = ∪ 𝐽
34	31, 33	sseqtrrdi 3988	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ℂ)
35	34	ad2antlr 727	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 ⊆ ℂ)
36		foimacnv 6817	. . . . . . 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 6031	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = (𝐺 “ ∪ 𝑡))
40		imauni 7220	. . . . . . 7 ⊢ (𝐺 “ ∪ 𝑡) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)
41	39, 40	eqtrdi 2780	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (𝐺 “ (^◡𝐺 “ 𝐴)) = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
42	37, 41	eqtr3d 2766	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → 𝐴 = ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤))
43	42	imaeq2d 6031	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)))
44		imaiun 7219	. . . 4 ⊢ (^◡𝐹 “ ∪ 𝑤 ∈ 𝑡 (𝐺 “ 𝑤)) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤))
45	43, 44	eqtrdi 2780	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) = ∪ 𝑤 ∈ 𝑡 (^◡𝐹 “ (𝐺 “ 𝑤)))
46		ssdomg 8971	. . . . . . 7 ⊢ (𝐾 ∈ TopBases → (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾))
47	23, 46	ax-mp 5	. . . . . 6 ⊢ (𝑡 ⊆ 𝐾 → 𝑡 ≼ 𝐾)
48		omelon 9599	. . . . . . . . . . 11 ⊢ ω ∈ On
49		nnenom 13945	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
50	49	ensymi 8975	. . . . . . . . . . 11 ⊢ ω ≈ ℕ
51		isnumi 9899	. . . . . . . . . . 11 ⊢ ((ω ∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom card)
52	48, 50, 51	mp2an 692	. . . . . . . . . 10 ⊢ ℕ ∈ dom card
53		qnnen 16181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ≈ ℕ
54		xpen 9104	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ))
55	53, 53, 54	mp2an 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ)
56		xpnnen 16179	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℕ × ℕ) ≈ ℕ
57	55, 56	entri 8979	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ≈ ℕ
58	57, 49	entr2i 8980	. . . . . . . . . . . . . . . . 17 ⊢ ω ≈ (ℚ × ℚ)
59		isnumi 9899	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card)
60	48, 58, 59	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (ℚ × ℚ) ∈ dom card
61		ioof 13408	. . . . . . . . . . . . . . . . . 18 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
62		ffun 6691	. . . . . . . . . . . . . . . . . 18 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
63	61, 62	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Fun (,)
64		qssre 12918	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℚ ⊆ ℝ
65		ressxr 11218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℝ^*
66	64, 65	sstri 3956	. . . . . . . . . . . . . . . . . . 19 ⊢ ℚ ⊆ ℝ^*
67		xpss12 5653	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℚ ⊆ ℝ^* ∧ ℚ ⊆ ℝ^) → (ℚ × ℚ) ⊆ (ℝ^ × ℝ^*))
68	66, 66, 67	mp2an 692	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ × ℚ) ⊆ (ℝ^* × ℝ^*)
69	61	fdmi 6699	. . . . . . . . . . . . . . . . . 18 ⊢ dom (,) = (ℝ^* × ℝ^*)
70	68, 69	sseqtrri 3996	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ × ℚ) ⊆ dom (,)
71		fores 6782	. . . . . . . . . . . . . . . . 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 10010	. . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . 14 ⊢ 𝐵 ≼ (ℚ × ℚ)
76		domentr 8984	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ℕ) → 𝐵 ≼ ℕ)
77	75, 57, 76	mp2an 692	. . . . . . . . . . . . 13 ⊢ 𝐵 ≼ ℕ
78	14	elexi 3470	. . . . . . . . . . . . . 14 ⊢ 𝐵 ∈ V
79	78	xpdom1 9040	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (𝐵 × 𝐵) ≼ (ℕ × 𝐵))
80	77, 79	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝐵 × 𝐵) ≼ (ℕ × 𝐵)
81		nnex 12192	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
82	81	xpdom2 9036	. . . . . . . . . . . . 13 ⊢ (𝐵 ≼ ℕ → (ℕ × 𝐵) ≼ (ℕ × ℕ))
83	77, 82	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℕ × 𝐵) ≼ (ℕ × ℕ)
84		domtr 8978	. . . . . . . . . . . 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 8984	. . . . . . . . . . 11 ⊢ (((𝐵 × 𝐵) ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → (𝐵 × 𝐵) ≼ ℕ)
87	85, 56, 86	mp2an 692	. . . . . . . . . 10 ⊢ (𝐵 × 𝐵) ≼ ℕ
88		numdom 9991	. . . . . . . . . 10 ⊢ ((ℕ ∈ dom card ∧ (𝐵 × 𝐵) ≼ ℕ) → (𝐵 × 𝐵) ∈ dom card)
89	52, 87, 88	mp2an 692	. . . . . . . . 9 ⊢ (𝐵 × 𝐵) ∈ dom card
90		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))
91		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
92		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
93	91, 92	xpex 7729	. . . . . . . . . . 11 ⊢ (𝑥 × 𝑦) ∈ V
94	90, 93	fnmpoi 8049	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)) Fn (𝐵 × 𝐵)
95		dffn4 6778	. . . . . . . . . 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 10010	. . . . . . . . 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 8978	. . . . . . . 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 5128	. . . . . 6 ⊢ 𝐾 ≼ ℕ
102		domtr 8978	. . . . . 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 2820	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐾 ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦)))
106	90, 93	elrnmpo 7525	. . . . . . . . 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 3930	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) ↔ (𝑧 ∈ (^◡(ℜ ∘ 𝐹) “ 𝑥) ∧ 𝑧 ∈ (^◡(ℑ ∘ 𝐹) “ 𝑦)))
109		mbff 25526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
110	109	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹:dom 𝐹⟶ℂ)
111		fvco3 6960	. . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (((ℜ ∘ 𝐹)‘𝑧) ∈ 𝑥 ↔ (ℜ‘(𝐹‘𝑧)) ∈ 𝑥))
114		fvco3 6960	. . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
119		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℜ‘𝑤) = (ℜ‘(𝐹‘𝑧)))
120		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = (𝐹‘𝑧) → (ℑ‘𝑤) = (ℑ‘(𝐹‘𝑧)))
121	119, 120	opeq12d 4845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = (𝐹‘𝑧) → ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩ = ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩)
122	1	cnrecnv 15131	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ^◡𝐺 = (𝑤 ∈ ℂ ↦ ⟨(ℜ‘𝑤), (ℑ‘𝑤)⟩)
123		opex 5424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ V
124	121, 122, 123	fvmpt 6968	. . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . 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 5674	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(ℜ‘(𝐹‘𝑧)), (ℑ‘(𝐹‘𝑧))⟩ ∈ (𝑥 × 𝑦) ↔ ((ℜ‘(𝐹‘𝑧)) ∈ 𝑥 ∧ (ℑ‘(𝐹‘𝑧)) ∈ 𝑦))
130		f1ocnv 6812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ))
131		f1ofn 6801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → ^◡𝐺 Fn ℂ)
132	28, 130, 131	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ ^◡𝐺 Fn ℂ
133		elpreima 7030	. . . . . . . . . . . . . . . . . . 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 6179	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡^◡𝐺 “ (𝑥 × 𝑦)) = (𝐺 “ (𝑥 × 𝑦))
136	135	eleq2i 2820	. . . . . . . . . . . . . . . . . 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 15078	. . . . . . . . . . . . . . . . . . 19 ⊢ ℜ:ℂ⟶ℝ
142		fco 6712	. . . . . . . . . . . . . . . . . . 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 6688	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℜ ∘ 𝐹):dom 𝐹⟶ℝ → (ℜ ∘ 𝐹) Fn dom 𝐹)
145		elpreima 7030	. . . . . . . . . . . . . . . . . 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 15079	. . . . . . . . . . . . . . . . . . 19 ⊢ ℑ:ℂ⟶ℝ
148		fco 6712	. . . . . . . . . . . . . . . . . . 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 6688	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ ∘ 𝐹):dom 𝐹⟶ℝ → (ℑ ∘ 𝐹) Fn dom 𝐹)
151		elpreima 7030	. . . . . . . . . . . . . . . . . 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 6688	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
158		elpreima 7030	. . . . . . . . . . . . . . . 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 2727	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((^◡(ℜ ∘ 𝐹) “ 𝑥) ∩ (^◡(ℑ ∘ 𝐹) “ 𝑦)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
164		ismbfcn 25530	. . . . . . . . . . . . . . . . . 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 25529	. . . . . . . . . . . . . . . 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 6042	. . . . . . . . . . . . . . 15 ⊢ ((,) “ (ℚ × ℚ)) ⊆ ran (,)
173	9, 172	eqsstri 3993	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ ran (,)
174		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
175	173, 174	sselid 3944	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ ran (,))
176		rsp 3225	. . . . . . . . . . . . 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 25529	. . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ ran (,))
185		rsp 3225	. . . . . . . . . . . . 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 25443	. . . . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ ((𝐹 ∈ MblFn ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))) ∈ dom vol)
190		imaeq2 6027	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑥 × 𝑦) → (𝐺 “ 𝑤) = (𝐺 “ (𝑥 × 𝑦)))
191	190	imaeq2d 6031	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 × 𝑦) → (^◡𝐹 “ (𝐺 “ 𝑤)) = (^◡𝐹 “ (𝐺 “ (𝑥 × 𝑦))))
192	191	eleq1d 2813	. . . . . . . . . 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 3194	. . . . . . . 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 3124	. . . . . 6 ⊢ (𝐹 ∈ MblFn → ∀𝑤 ∈ 𝐾 (^◡𝐹 “ (𝐺 “ 𝑤)) ∈ dom vol)
197		ssralv 4015	. . . . . 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 25458	. . . 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 2828	. 2 ⊢ (((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) ∧ (𝑡 ⊆ 𝐾 ∧ (^◡𝐺 “ 𝐴) = ∪ 𝑡)) → (^◡𝐹 “ 𝐴) ∈ dom vol)
203	27, 202	exlimddv 1935	1 ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4563 ⟨cop 4595 ∪ cuni 4871 ∪ ciun 4955 class class class wbr 5107 × cxp 5636 ^◡ccnv 5637 dom cdm 5638 ran crn 5639 ↾ cres 5640 “ cima 5641 ∘ ccom 5642 Oncon0 6332 Fun wfun 6505 Fn wfn 6506 ⟶wf 6507 –onto→wfo 6509 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ωcom 7842 ≈ cen 8915 ≼ cdom 8916 cardccrd 9888 ℂcc 11066 ℝcr 11067 ici 11070 + caddc 11071 · cmul 11073 ℝ^*cxr 11207 ℕcn 12186 ℚcq 12907 (,)cioo 13306 ℜcre 15063 ℑcim 15064 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 TopBasesctb 22832 Cn ccn 23111 ×_t ctx 23447 Homeochmeo 23640 volcvol 25364 MblFncmbf 25515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cn 23114 df-cnp 23115 df-tx 23449 df-hmeo 23642 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-ovol 25365 df-vol 25366 df-mbf 25520

This theorem is referenced by: mbfimaopn 25557

W3C validator