Theorem smfpimcc 43880

Description: Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function ℎ that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of 𝐴. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smfpimcc.1	⊢ Ⅎ𝑛𝐹
smfpimcc.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smfpimcc.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpimcc.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfpimcc.j	⊢ 𝐽 = (topGen‘ran (,))
smfpimcc.b	⊢ 𝐵 = (SalGen‘𝐽)
smfpimcc.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Assertion

Ref	Expression
smfpimcc	⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))

Distinct variable groups:   𝐴,ℎ,𝑛   ℎ,𝐹   𝑆,ℎ   ℎ,𝑍,𝑛

Allowed substitution hints:   𝜑(ℎ,𝑛)   𝐵(ℎ,𝑛)   𝑆(𝑛)   𝐹(𝑛)   𝐽(ℎ,𝑛)   𝑀(ℎ,𝑛)

Proof of Theorem smfpimcc

Dummy variables 𝑓 𝑚 𝑠 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpimcc.z	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
2	1	uzct 42149	. . . . . 6 ⊢ 𝑍 ≼ ω
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
4		mptct 10038	. . . . 5 ⊢ (𝑍 ≼ ω → (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
5		rnct 10025	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω → ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝜑 → ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
7		vex 3402	. . . . . . . 8 ⊢ 𝑦 ∈ V
8		eqid 2738	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) = (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
9	8	elrnmpt 5799	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ↔ ∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}))
10	7, 9	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ↔ ∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
11	10	biimpi 219	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) → ∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
12	11	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) → ∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
13		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
14		smfpimcc.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
15	14	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
16		smfpimcc.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
17	16	ffvelrnda 6861	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
18		eqid 2738	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
19		smfpimcc.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = (topGen‘ran (,))
20		smfpimcc.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (SalGen‘𝐽)
21		smfpimcc.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
22	21	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝐵)
23		eqid 2738	. . . . . . . . . . . . 13 ⊢ (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑚) “ 𝐴)
24	15, 17, 18, 19, 20, 22, 23	smfpimbor1 43873	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
25		fvex 6687	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑚) ∈ V
26	25	dmex 7642	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑚) ∈ V
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
28		elrest 16804	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ SAlg ∧ dom (𝐹‘𝑚) ∈ V) → ((◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))))
29	14, 27, 28	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))))
30	29	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))))
31	24, 30	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
32		rabn0 4274	. . . . . . . . . . 11 ⊢ ({𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅ ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
33	31, 32	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
34	33	3adant3 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
35	13, 34	eqnetrd 3001	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 ≠ ∅)
36	35	3exp 1120	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑍 → (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅)))
37	36	rexlimdv 3193	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
38	37	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) → (∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
39	12, 38	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) → 𝑦 ≠ ∅)
40	6, 39	axccd2 42307	. . 3 ⊢ (𝜑 → ∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
41		nfv 1921	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
42		nfmpt1 5128	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
43	42	nfrn 5795	. . . . . . . 8 ⊢ Ⅎ𝑚ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
44		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑓‘𝑦) ∈ 𝑦
45	43, 44	nfralw 3138	. . . . . . 7 ⊢ Ⅎ𝑚∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦
46	41, 45	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
47	1	fvexi 6688	. . . . . 6 ⊢ 𝑍 ∈ V
48	14	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
49		fveq2 6674	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
50		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
51	49, 50	eleq12d 2827	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑤) ∈ 𝑤))
52	51	rspccva 3525	. . . . . . 7 ⊢ ((∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦 ∧ 𝑤 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) → (𝑓‘𝑤) ∈ 𝑤)
53	52	adantll 714	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦) ∧ 𝑤 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) → (𝑓‘𝑤) ∈ 𝑤)
54		eqid 2738	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑓‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})) = (𝑚 ∈ 𝑍 ↦ (𝑓‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}))
55	46, 47, 48, 53, 54	smfpimcclem 43879	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦) → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))))
56	55	ex 416	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)))))
57	56	exlimdv 1940	. . 3 ⊢ (𝜑 → (∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)))))
58	40, 57	mpd 15	. 2 ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))))
59		smfpimcc.1	. . . . . . . . 9 ⊢ Ⅎ𝑛𝐹
60		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑚
61	59, 60	nffv 6684	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐹‘𝑚)
62	61	nfcnv 5721	. . . . . . 7 ⊢ Ⅎ𝑛◡(𝐹‘𝑚)
63		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
64	62, 63	nfima 5911	. . . . . 6 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴)
65		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑛(ℎ‘𝑚)
66	61	nfdm 5794	. . . . . . 7 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
67	65, 66	nfin 4107	. . . . . 6 ⊢ Ⅎ𝑛((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
68	64, 67	nfeq 2912	. . . . 5 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
69		nfv 1921	. . . . 5 ⊢ Ⅎ𝑚(◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))
70		fveq2 6674	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
71	70	cnveqd 5718	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ◡(𝐹‘𝑚) = ◡(𝐹‘𝑛))
72	71	imaeq1d 5902	. . . . . 6 ⊢ (𝑚 = 𝑛 → (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑛) “ 𝐴))
73		fveq2 6674	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (ℎ‘𝑚) = (ℎ‘𝑛))
74	70	dmeqd 5748	. . . . . . 7 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
75	73, 74	ineq12d 4104	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
76	72, 75	eqeq12d 2754	. . . . 5 ⊢ (𝑚 = 𝑛 → ((◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) ↔ (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
77	68, 69, 76	cbvralw 3340	. . . 4 ⊢ (∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) ↔ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
78	77	anbi2i 626	. . 3 ⊢ ((ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))) ↔ (ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
79	78	exbii 1854	. 2 ⊢ (∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))) ↔ ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
80	58, 79	sylib 221	1 ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542  ∃wex 1786   ∈ wcel 2114  Ⅎwnfc 2879   ≠ wne 2934  ∀wral 3053  ∃wrex 3054  {crab 3057  Vcvv 3398   ∩ cin 3842  ∅c0 4211   class class class wbr 5030   ↦ cmpt 5110  ^◡ccnv 5524  dom cdm 5525  ran crn 5526   “ cima 5528  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  ωcom 7599   ≼ cdom 8553  ℤ_≥cuz 12324  (,)cioo 12821   ↾_t crest 16797  topGenctg 16814  SAlgcsalg 43391  SalGencsalgen 43395  SMblFncsmblfn 43775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cc 9935  ax-ac2 9963  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-oadd 8135  df-omul 8136  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-oi 9047  df-card 9441  df-acn 9444  df-ac 9616  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-q 12431  df-rp 12473  df-ioo 12825  df-ico 12827  df-fl 13253  df-rest 16799  df-topgen 16820  df-top 21645  df-bases 21697  df-salg 43392  df-salgen 43396  df-smblfn 43776

This theorem is referenced by:  smfsuplem2  43884