imambfm - Mathbox for Thierry Arnoux

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Theorem imambfm 32129

Description: If the sigma-algebra in the range of a given function is generated by a collection of basic sets 𝐾, then to check the measurability of that function, we need only consider inverse images of basic sets 𝑎. (Contributed by Thierry Arnoux, 4-Jun-2017.)

**Hypotheses**
Ref	Expression
imambfm.1	⊢ (𝜑 → 𝐾 ∈ V)
imambfm.2	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
imambfm.3	⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾))

**Assertion**
Ref	Expression
imambfm	⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)))

Distinct variable groups: 𝐹,𝑎 𝐾,𝑎 𝑆,𝑎 𝑇,𝑎 𝜑,𝑎

Proof of Theorem imambfm Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imambfm.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → 𝑆 ∈ ∪ ran sigAlgebra)
3		imambfm.3	. . . . . 6 ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾))
4		imambfm.1	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V)
5	4	sgsiga 32010	. . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐾) ∈ ∪ ran sigAlgebra)
6	3, 5	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → 𝑇 ∈ ∪ ran sigAlgebra)
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → 𝐹 ∈ (𝑆MblFnM𝑇))
9	2, 7, 8	mbfmf 32122	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
10	1	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑆 ∈ ∪ ran sigAlgebra)
11	6	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑇 ∈ ∪ ran sigAlgebra)
12		simplr 765	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝐹 ∈ (𝑆MblFnM𝑇))
13		sssigagen 32013	. . . . . . . . 9 ⊢ (𝐾 ∈ V → 𝐾 ⊆ (sigaGen‘𝐾))
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ (sigaGen‘𝐾))
15	14, 3	sseqtrrd 3958	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑇)
16	15	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝐾 ⊆ 𝑇)
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑎 ∈ 𝐾)
18	16, 17	sseldd 3918	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑎 ∈ 𝑇)
19	10, 11, 12, 18	mbfmcnvima 32124	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → (^◡𝐹 “ 𝑎) ∈ 𝑆)
20	19	ralrimiva 3107	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)
21	9, 20	jca 511	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆))
22		unielsiga 31996	. . . . . 6 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∪ 𝑇 ∈ 𝑇)
23	6, 22	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑇)
24	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∪ 𝑇 ∈ 𝑇)
25		unielsiga 31996	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∪ 𝑆 ∈ 𝑆)
26	1, 25	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∪ 𝑆 ∈ 𝑆)
28		simprl 767	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
29		elmapg 8586	. . . . 5 ⊢ ((∪ 𝑇 ∈ 𝑇 ∧ ∪ 𝑆 ∈ 𝑆) → (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ↔ 𝐹:∪ 𝑆⟶∪ 𝑇))
30	29	biimpar 477	. . . 4 ⊢ (((∪ 𝑇 ∈ 𝑇 ∧ ∪ 𝑆 ∈ 𝑆) ∧ 𝐹:∪ 𝑆⟶∪ 𝑇) → 𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆))
31	24, 27, 28, 30	syl21anc 834	. . 3 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆))
32	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑇 = (sigaGen‘𝐾))
33		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝜑)
34		ssrab2 4009	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝑇
35		pwuni 4875	. . . . . . . . . . 11 ⊢ 𝑇 ⊆ 𝒫 ∪ 𝑇
36	34, 35	sstri 3926	. . . . . . . . . 10 ⊢ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 ∪ 𝑇
37	36	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 ∪ 𝑇)
38		fimacnv 6606	. . . . . . . . . . . . 13 ⊢ (𝐹:∪ 𝑆⟶∪ 𝑇 → (^◡𝐹 “ ∪ 𝑇) = ∪ 𝑆)
39	38	ad2antrl 724	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → (^◡𝐹 “ ∪ 𝑇) = ∪ 𝑆)
40	39, 27	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → (^◡𝐹 “ ∪ 𝑇) ∈ 𝑆)
41		imaeq2 5954	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑇 → (^◡𝐹 “ 𝑎) = (^◡𝐹 “ ∪ 𝑇))
42	41	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑇 → ((^◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (^◡𝐹 “ ∪ 𝑇) ∈ 𝑆))
43	42	elrab 3617	. . . . . . . . . . 11 ⊢ (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (∪ 𝑇 ∈ 𝑇 ∧ (^◡𝐹 “ ∪ 𝑇) ∈ 𝑆))
44	24, 40, 43	sylanbrc 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
45	6	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝑇 ∈ ∪ ran sigAlgebra)
46	45, 22	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → ∪ 𝑇 ∈ 𝑇)
47		elrabi 3611	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} → 𝑥 ∈ 𝑇)
48	47	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝑥 ∈ 𝑇)
49		difelsiga 32001	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑇 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
50	45, 46, 48, 49	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
51		simplrl 773	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝐹:∪ 𝑆⟶∪ 𝑇)
52		ffun 6587	. . . . . . . . . . . . . 14 ⊢ (𝐹:∪ 𝑆⟶∪ 𝑇 → Fun 𝐹)
53		difpreima 6924	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) = ((^◡𝐹 “ ∪ 𝑇) ∖ (^◡𝐹 “ 𝑥)))
54	51, 52, 53	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) = ((^◡𝐹 “ ∪ 𝑇) ∖ (^◡𝐹 “ 𝑥)))
55	39	difeq1d 4052	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ((^◡𝐹 “ ∪ 𝑇) ∖ (^◡𝐹 “ 𝑥)) = (∪ 𝑆 ∖ (^◡𝐹 “ 𝑥)))
56	55	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → ((^◡𝐹 “ ∪ 𝑇) ∖ (^◡𝐹 “ 𝑥)) = (∪ 𝑆 ∖ (^◡𝐹 “ 𝑥)))
57	1	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝑆 ∈ ∪ ran sigAlgebra)
58	57, 25	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → ∪ 𝑆 ∈ 𝑆)
59		imaeq2 5954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → (^◡𝐹 “ 𝑎) = (^◡𝐹 “ 𝑥))
60	59	eleq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑥 → ((^◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (^◡𝐹 “ 𝑥) ∈ 𝑆))
61	60	elrab 3617	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (𝑥 ∈ 𝑇 ∧ (^◡𝐹 “ 𝑥) ∈ 𝑆))
62	61	simprbi 496	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} → (^◡𝐹 “ 𝑥) ∈ 𝑆)
63	62	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (^◡𝐹 “ 𝑥) ∈ 𝑆)
64		difelsiga 32001	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 ∈ 𝑆 ∧ (^◡𝐹 “ 𝑥) ∈ 𝑆) → (∪ 𝑆 ∖ (^◡𝐹 “ 𝑥)) ∈ 𝑆)
65	57, 58, 63, 64	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (∪ 𝑆 ∖ (^◡𝐹 “ 𝑥)) ∈ 𝑆)
66	56, 65	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → ((^◡𝐹 “ ∪ 𝑇) ∖ (^◡𝐹 “ 𝑥)) ∈ 𝑆)
67	54, 66	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆)
68		imaeq2 5954	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (∪ 𝑇 ∖ 𝑥) → (^◡𝐹 “ 𝑎) = (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)))
69	68	eleq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑎 = (∪ 𝑇 ∖ 𝑥) → ((^◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆))
70	69	elrab 3617	. . . . . . . . . . . 12 ⊢ ((∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ ((∪ 𝑇 ∖ 𝑥) ∈ 𝑇 ∧ (^◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆))
71	50, 67, 70	sylanbrc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
72	71	ralrimiva 3107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∀𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
73	6	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → 𝑇 ∈ ∪ ran sigAlgebra)
74	34	sspwi 4544	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 𝑇
75	74	sseli 3913	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} → 𝑥 ∈ 𝒫 𝑇)
76	75	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → 𝑥 ∈ 𝒫 𝑇)
77		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω)
78		sigaclcu 31985	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
79	73, 76, 77, 78	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
80		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆))
81	80	simpld 494	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → 𝐹:∪ 𝑆⟶∪ 𝑇)
82		unipreima 30882	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (^◡𝐹 “ ∪ 𝑥) = ∪ 𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦))
83	81, 52, 82	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → (^◡𝐹 “ ∪ 𝑥) = ∪ 𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦))
84	1	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → 𝑆 ∈ ∪ ran sigAlgebra)
85		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
86		simpllr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
87		elelpwi 4542	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝑦 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
88	85, 86, 87	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
89		imaeq2 5954	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑦 → (^◡𝐹 “ 𝑎) = (^◡𝐹 “ 𝑦))
90	89	eleq1d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑦 → ((^◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (^◡𝐹 “ 𝑦) ∈ 𝑆))
91	90	elrab 3617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (𝑦 ∈ 𝑇 ∧ (^◡𝐹 “ 𝑦) ∈ 𝑆))
92	91	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} → (^◡𝐹 “ 𝑦) ∈ 𝑆)
93	88, 92	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) ∧ 𝑦 ∈ 𝑥) → (^◡𝐹 “ 𝑦) ∈ 𝑆)
94	93	ralrimiva 3107	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → ∀𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦) ∈ 𝑆)
95		nfcv 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝑥
96	95	sigaclcuni 31986	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦) ∈ 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦) ∈ 𝑆)
97	84, 94, 77, 96	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 (^◡𝐹 “ 𝑦) ∈ 𝑆)
98	83, 97	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → (^◡𝐹 “ ∪ 𝑥) ∈ 𝑆)
99		imaeq2 5954	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ∪ 𝑥 → (^◡𝐹 “ 𝑎) = (^◡𝐹 “ ∪ 𝑥))
100	99	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∪ 𝑥 → ((^◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (^◡𝐹 “ ∪ 𝑥) ∈ 𝑆))
101	100	elrab 3617	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (∪ 𝑥 ∈ 𝑇 ∧ (^◡𝐹 “ ∪ 𝑥) ∈ 𝑆))
102	79, 98, 101	sylanbrc 582	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
103	102	ex 412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}))
104	103	ralrimiva 3107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∀𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}))
105	44, 72, 104	3jca 1126	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})))
106		rabexg 5250	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ V)
107		issiga 31980	. . . . . . . . . . 11 ⊢ ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ V → ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇) ↔ ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 ∪ 𝑇 ∧ (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})))))
108	6, 106, 107	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇) ↔ ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 ∪ 𝑇 ∧ (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})))))
109	108	biimpar 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 ∪ 𝑇 ∧ (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∧ ∀𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} (𝑥 ≼ ω → ∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})))) → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇))
110	33, 37, 105, 109	syl12anc 833	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇))
111	3	unieqd 4850	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑇 = ∪ (sigaGen‘𝐾))
112		unisg 32011	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ V → ∪ (sigaGen‘𝐾) = ∪ 𝐾)
113	4, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (sigaGen‘𝐾) = ∪ 𝐾)
114	111, 113	eqtrd 2778	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑇 = ∪ 𝐾)
115	114	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝜑 → (sigAlgebra‘∪ 𝑇) = (sigAlgebra‘∪ 𝐾))
116	115	eleq2d 2824	. . . . . . . . 9 ⊢ (𝜑 → ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇) ↔ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝐾)))
117	116	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇) ↔ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝐾)))
118	110, 117	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝐾))
119	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐾 ⊆ 𝑇)
120		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)
121		ssrab 4002	. . . . . . . 8 ⊢ (𝐾 ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (𝐾 ⊆ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆))
122	119, 120, 121	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐾 ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
123		sigagenss 32017	. . . . . . 7 ⊢ (({𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝐾) ∧ 𝐾 ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆}) → (sigaGen‘𝐾) ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
124	118, 122, 123	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → (sigaGen‘𝐾) ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
125	32, 124	eqsstrd 3955	. . . . 5 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑇 ⊆ {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
126	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝑇)
127	125, 126	eqssd 3934	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑇 = {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆})
128		rabid2 3307	. . . 4 ⊢ (𝑇 = {𝑎 ∈ 𝑇 ∣ (^◡𝐹 “ 𝑎) ∈ 𝑆} ↔ ∀𝑎 ∈ 𝑇 (^◡𝐹 “ 𝑎) ∈ 𝑆)
129	127, 128	sylib 217	. . 3 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → ∀𝑎 ∈ 𝑇 (^◡𝐹 “ 𝑎) ∈ 𝑆)
130	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑆 ∈ ∪ ran sigAlgebra)
131	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑇 ∈ ∪ ran sigAlgebra)
132	130, 131	ismbfm 32119	. . 3 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑_m ∪ 𝑆) ∧ ∀𝑎 ∈ 𝑇 (^◡𝐹 “ 𝑎) ∈ 𝑆)))
133	31, 129, 132	mpbir2and 709	. 2 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐹 ∈ (𝑆MblFnM𝑇))
134	21, 133	impbida 797	1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (^◡𝐹 “ 𝑎) ∈ 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 𝒫 cpw 4530 ∪ cuni 4836 ∪ ciun 4921 class class class wbr 5070 ^◡ccnv 5579 ran crn 5581 “ cima 5583 Fun wfun 6412 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ωcom 7687 ↑_m cmap 8573 ≼ cdom 8689 sigAlgebracsiga 31976 sigaGencsigagen 32006 MblFnMcmbfm 32117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-ac2 10150

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-siga 31977 df-sigagen 32007 df-mbfm 32118

This theorem is referenced by: cnmbfm 32130 mbfmco2 32132

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