Theorem imambfm 34246

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 34125	. . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐾) ∈ ∪ ran sigAlgebra)
6	3, 5	eqeltrd 2828	. . . . 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 34237	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
10	1	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑆 ∈ ∪ ran sigAlgebra)
11	6	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑇 ∈ ∪ ran sigAlgebra)
12		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝐹 ∈ (𝑆MblFnM𝑇))
13		sssigagen 34128	. . . . . . . . 9 ⊢ (𝐾 ∈ V → 𝐾 ⊆ (sigaGen‘𝐾))
14	4, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ⊆ (sigaGen‘𝐾))
15	14, 3	sseqtrrd 3981	. . . . . . 7 ⊢ (𝜑 → 𝐾 ⊆ 𝑇)
16	15	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝐾 ⊆ 𝑇)
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑎 ∈ 𝐾)
18	16, 17	sseldd 3944	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → 𝑎 ∈ 𝑇)
19	10, 11, 12, 18	mbfmcnvima 34239	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ (𝑆MblFnM𝑇)) ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝑆)
20	19	ralrimiva 3125	. . 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 34111	. . . . . 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 34111	. . . . . 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐹:∪ 𝑆⟶∪ 𝑇)
29		elmapg 8789	. . . . 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 837	. . 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 4039	. . . . . . . . . . 11 ⊢ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝑇
35		pwuni 4905	. . . . . . . . . . 11 ⊢ 𝑇 ⊆ 𝒫 ∪ 𝑇
36	34, 35	sstri 3953	. . . . . . . . . 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 6692	. . . . . . . . . . . . 13 ⊢ (𝐹:∪ 𝑆⟶∪ 𝑇 → (◡𝐹 “ ∪ 𝑇) = ∪ 𝑆)
39	38	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → (◡𝐹 “ ∪ 𝑇) = ∪ 𝑆)
40	39, 27	eqeltrd 2828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → (◡𝐹 “ ∪ 𝑇) ∈ 𝑆)
41		imaeq2 6016	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑇 → (◡𝐹 “ 𝑎) = (◡𝐹 “ ∪ 𝑇))
42	41	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑇 → ((◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (◡𝐹 “ ∪ 𝑇) ∈ 𝑆))
43	42	elrab 3656	. . . . . . . . . . 11 ⊢ (∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (∪ 𝑇 ∈ 𝑇 ∧ (◡𝐹 “ ∪ 𝑇) ∈ 𝑆))
44	24, 40, 43	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → ∪ 𝑇 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆})
45	6	ad2antrr 726	. . . . . . . . . . . . 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 3651	. . . . . . . . . . . . . 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 34116	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑇 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
50	45, 46, 48, 49	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) → (∪ 𝑇 ∖ 𝑥) ∈ 𝑇)
51		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝐹:∪ 𝑆⟶∪ 𝑇)
52		ffun 6673	. . . . . . . . . . . . . 14 ⊢ (𝐹:∪ 𝑆⟶∪ 𝑇 → Fun 𝐹)
53		difpreima 7019	. . . . . . . . . . . . . 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 4084	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 6016	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑥))
60	59	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑥 → ((◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆))
61	60	elrab 3656	. . . . . . . . . . . . . . . . 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 34116	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 ∈ 𝑆 ∧ (◡𝐹 “ 𝑥) ∈ 𝑆) → (∪ 𝑆 ∖ (◡𝐹 “ 𝑥)) ∈ 𝑆)
65	57, 58, 63, 64	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) → (∪ 𝑆 ∖ (◡𝐹 “ 𝑥)) ∈ 𝑆)
66	56, 65	eqeltrd 2828	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) → (◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆)
68		imaeq2 6016	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (∪ 𝑇 ∖ 𝑥) → (◡𝐹 “ 𝑎) = (◡𝐹 “ (∪ 𝑇 ∖ 𝑥)))
69	68	eleq1d 2813	. . . . . . . . . . . . 13 ⊢ (𝑎 = (∪ 𝑇 ∖ 𝑥) → ((◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆))
70	69	elrab 3656	. . . . . . . . . . . 12 ⊢ ((∪ 𝑇 ∖ 𝑥) ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ↔ ((∪ 𝑇 ∖ 𝑥) ∈ 𝑇 ∧ (◡𝐹 “ (∪ 𝑇 ∖ 𝑥)) ∈ 𝑆))
71	50, 67, 70	sylanbrc 583	. . . . . . . . . . 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 3125	. . . . . . . . . 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 730	. . . . . . . . . . . . . 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 4571	. . . . . . . . . . . . . . . 16 ⊢ 𝒫 {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ⊆ 𝒫 𝑇
75	74	sseli 3939	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} → 𝑥 ∈ 𝒫 𝑇)
76	75	ad2antlr 727	. . . . . . . . . . . . . 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 34100	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
79	73, 76, 77, 78	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
80		simpllr 775	. . . . . . . . . . . . . . . 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 32617	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . 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 4569	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) → 𝑦 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆})
88	85, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . 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 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑦 → (◡𝐹 “ 𝑎) = (◡𝐹 “ 𝑦))
90	89	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑦 → ((◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (◡𝐹 “ 𝑦) ∈ 𝑆))
91	90	elrab 3656	. . . . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . . . 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 2891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝑥
96	95	sigaclcuni 34101	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑦 ∈ 𝑥 (◡𝐹 “ 𝑦) ∈ 𝑆 ∧ 𝑥 ≼ ω) → ∪ 𝑦 ∈ 𝑥 (◡𝐹 “ 𝑦) ∈ 𝑆)
97	84, 94, 77, 96	syl3anc 1373	. . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) ∧ 𝑥 ∈ 𝒫 {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆}) ∧ 𝑥 ≼ ω) → (◡𝐹 “ ∪ 𝑥) ∈ 𝑆)
99		imaeq2 6016	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ∪ 𝑥 → (◡𝐹 “ 𝑎) = (◡𝐹 “ ∪ 𝑥))
100	99	eleq1d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∪ 𝑥 → ((◡𝐹 “ 𝑎) ∈ 𝑆 ↔ (◡𝐹 “ ∪ 𝑥) ∈ 𝑆))
101	100	elrab 3656	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (∪ 𝑥 ∈ 𝑇 ∧ (◡𝐹 “ ∪ 𝑥) ∈ 𝑆))
102	79, 98, 101	sylanbrc 583	. . . . . . . . . . . 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 3125	. . . . . . . . . 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 1128	. . . . . . . . 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 5287	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ∈ V)
107		issiga 34095	. . . . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ∈ (sigAlgebra‘∪ 𝑇))
111	3	unieqd 4880	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑇 = ∪ (sigaGen‘𝐾))
112		unisg 34126	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ V → ∪ (sigaGen‘𝐾) = ∪ 𝐾)
113	4, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ (sigaGen‘𝐾) = ∪ 𝐾)
114	111, 113	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑇 = ∪ 𝐾)
115	114	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝜑 → (sigAlgebra‘∪ 𝑇) = (sigAlgebra‘∪ 𝐾))
116	115	eleq2d 2814	. . . . . . . . 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 232	. . . . . . 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 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)
121		ssrab 4032	. . . . . . . 8 ⊢ (𝐾 ⊆ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ↔ (𝐾 ⊆ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))
122	119, 120, 121	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐾 ⊆ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆})
123		sigagenss 34132	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → (sigaGen‘𝐾) ⊆ {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆})
125	32, 124	eqsstrd 3978	. . . . 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 3961	. . . 4 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝑇 = {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆})
128		rabid2 3436	. . . 4 ⊢ (𝑇 = {𝑎 ∈ 𝑇 ∣ (◡𝐹 “ 𝑎) ∈ 𝑆} ↔ ∀𝑎 ∈ 𝑇 (◡𝐹 “ 𝑎) ∈ 𝑆)
129	127, 128	sylib 218	. . 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 34234	. . 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 713	. 2 ⊢ ((𝜑 ∧ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)) → 𝐹 ∈ (𝑆MblFnM𝑇))
134	21, 133	impbida 800	1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  Vcvv 3444   ∖ cdif 3908   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5102  ^◡ccnv 5630  ran crn 5632   “ cima 5634  Fun wfun 6493  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ωcom 7822   ↑_m cmap 8776   ≼ cdom 8893  sigAlgebracsiga 34091  sigaGencsigagen 34121  MblFnMcmbfm 34232

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-ac2 10392

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-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9439  df-dju 9830  df-card 9868  df-acn 9871  df-ac 10045  df-siga 34092  df-sigagen 34122  df-mbfm 34233

This theorem is referenced by:  cnmbfm  34247  mbfmco2  34249