Theorem smfpimcc 47254

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 45512	. . . . . 6 ⊢ 𝑍 ≼ ω
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
4		mptct 10451	. . . . 5 ⊢ (𝑍 ≼ ω → (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
5		rnct 10438	. . . . 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 3434	. . . . . . . 8 ⊢ 𝑦 ∈ V
8		eqid 2737	. . . . . . . . 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 5907	. . . . . . . 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 216	. . . . . 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 481	. . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
16		smfpimcc.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
17	16	ffvelcdmda 7030	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
18		eqid 2737	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝐵)
23		eqid 2737	. . . . . . . . . . . . 13 ⊢ (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑚) “ 𝐴)
24	15, 17, 18, 19, 20, 22, 23	smfpimbor1 47246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
25		fvex 6847	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑚) ∈ V
26	25	dmex 7853	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑚) ∈ V
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
28		elrest 17381	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))))
30	29	adantr 480	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
32		rabn0 4330	. . . . . . . . . . 11 ⊢ ({𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅ ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
33	31, 32	sylibr 234	. . . . . . . . . 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 3000	. . . . . . . 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 3137	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
38	37	adantr 480	. . . . 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 45677	. . 3 ⊢ (𝜑 → ∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
41		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
42		nfmpt1 5185	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
43	42	nfrn 5901	. . . . . . . 8 ⊢ Ⅎ𝑚ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
44		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑓‘𝑦) ∈ 𝑦
45	43, 44	nfralw 3285	. . . . . . 7 ⊢ Ⅎ𝑚∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦
46	41, 45	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
47	1	fvexi 6848	. . . . . 6 ⊢ 𝑍 ∈ V
48	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
49		fveq2 6834	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
50		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
51	49, 50	eleq12d 2831	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑤) ∈ 𝑤))
52	51	rspccva 3564	. . . . . . 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 715	. . . . . 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 2737	. . . . . 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 47253	. . . . 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 412	. . . 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 1935	. . 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 6844	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐹‘𝑚)
62	61	nfcnv 5827	. . . . . . 7 ⊢ Ⅎ𝑛◡(𝐹‘𝑚)
63		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
64	62, 63	nfima 6027	. . . . . 6 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴)
65		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑛(ℎ‘𝑚)
66	61	nfdm 5900	. . . . . . 7 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
67	65, 66	nfin 4165	. . . . . 6 ⊢ Ⅎ𝑛((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
68	64, 67	nfeq 2913	. . . . 5 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
69		nfv 1916	. . . . 5 ⊢ Ⅎ𝑚(◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))
70		fveq2 6834	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
71	70	cnveqd 5824	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ◡(𝐹‘𝑚) = ◡(𝐹‘𝑛))
72	71	imaeq1d 6018	. . . . . 6 ⊢ (𝑚 = 𝑛 → (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑛) “ 𝐴))
73		fveq2 6834	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (ℎ‘𝑚) = (ℎ‘𝑛))
74	70	dmeqd 5854	. . . . . . 7 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
75	73, 74	ineq12d 4162	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
76	72, 75	eqeq12d 2753	. . . . 5 ⊢ (𝑚 = 𝑛 → ((◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) ↔ (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
77	68, 69, 76	cbvralw 3280	. . . 4 ⊢ (∀𝑚 ∈ 𝑍 (◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) ↔ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
78	77	anbi2i 624	. . 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 1850	. 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 218	1 ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∩ cin 3889  ∅c0 4274   class class class wbr 5086   ↦ cmpt 5167  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ωcom 7810   ≼ cdom 8884  ℤ_≥cuz 12779  (,)cioo 13289   ↾_t crest 17374  topGenctg 17391  SAlgcsalg 46754  SalGencsalgen 46758  SMblFncsmblfn 47141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cc 10348  ax-ac2 10376  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-omul 8403  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-acn 9857  df-ac 10029  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-ioo 13293  df-ico 13295  df-fl 13742  df-rest 17376  df-topgen 17397  df-top 22869  df-bases 22921  df-salg 46755  df-salgen 46759  df-smblfn 47142

This theorem is referenced by:  smfsuplem2  47258