Theorem smfpimcc 45459

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 43683	. . . . . 6 ⊢ 𝑍 ≼ ω
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
4		mptct 10529	. . . . 5 ⊢ (𝑍 ≼ ω → (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
5		rnct 10516	. . . . 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 3479	. . . . . . . 8 ⊢ 𝑦 ∈ V
8		eqid 2733	. . . . . . . . 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 5953	. . . . . . . 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 215	. . . . . 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 483	. . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
16		smfpimcc.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
17	16	ffvelcdmda 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
18		eqid 2733	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝐵)
23		eqid 2733	. . . . . . . . . . . . 13 ⊢ (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑚) “ 𝐴)
24	15, 17, 18, 19, 20, 22, 23	smfpimbor1 45451	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
25		fvex 6901	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑚) ∈ V
26	25	dmex 7897	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑚) ∈ V
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
28		elrest 17369	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
32		rabn0 4384	. . . . . . . . . . 11 ⊢ ({𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅ ↔ ∃𝑠 ∈ 𝑆 (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚)))
33	31, 32	sylibr 233	. . . . . . . . . 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 3009	. . . . . . . 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 3154	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
38	37	adantr 482	. . . . 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 43862	. . 3 ⊢ (𝜑 → ∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
41		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
42		nfmpt1 5255	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
43	42	nfrn 5949	. . . . . . . 8 ⊢ Ⅎ𝑚ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
44		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑓‘𝑦) ∈ 𝑦
45	43, 44	nfralw 3309	. . . . . . 7 ⊢ Ⅎ𝑚∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦
46	41, 45	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
47	1	fvexi 6902	. . . . . 6 ⊢ 𝑍 ∈ V
48	14	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
49		fveq2 6888	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
50		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
51	49, 50	eleq12d 2828	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑤) ∈ 𝑤))
52	51	rspccva 3611	. . . . . . 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 713	. . . . . 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 2733	. . . . . 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 45458	. . . . 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 414	. . . 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 1937	. . 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 2904	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑚
61	59, 60	nffv 6898	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐹‘𝑚)
62	61	nfcnv 5876	. . . . . . 7 ⊢ Ⅎ𝑛◡(𝐹‘𝑚)
63		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
64	62, 63	nfima 6065	. . . . . 6 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴)
65		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑛(ℎ‘𝑚)
66	61	nfdm 5948	. . . . . . 7 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
67	65, 66	nfin 4215	. . . . . 6 ⊢ Ⅎ𝑛((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
68	64, 67	nfeq 2917	. . . . 5 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
69		nfv 1918	. . . . 5 ⊢ Ⅎ𝑚(◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))
70		fveq2 6888	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
71	70	cnveqd 5873	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ◡(𝐹‘𝑚) = ◡(𝐹‘𝑛))
72	71	imaeq1d 6056	. . . . . 6 ⊢ (𝑚 = 𝑛 → (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑛) “ 𝐴))
73		fveq2 6888	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (ℎ‘𝑚) = (ℎ‘𝑛))
74	70	dmeqd 5903	. . . . . . 7 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
75	73, 74	ineq12d 4212	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
76	72, 75	eqeq12d 2749	. . . . 5 ⊢ (𝑚 = 𝑛 → ((◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) ↔ (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
77	68, 69, 76	cbvralw 3304	. . . 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 1851	. 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 217	1 ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  Ⅎwnfc 2884   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ∩ cin 3946  ∅c0 4321   class class class wbr 5147   ↦ cmpt 5230  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   “ cima 5678  ⟶wf 6536  ‘cfv 6540  (class class class)co 7404  ωcom 7850   ≼ cdom 8933  ℤ_≥cuz 12818  (,)cioo 13320   ↾_t crest 17362  topGenctg 17379  SAlgcsalg 44959  SalGencsalgen 44963  SMblFncsmblfn 45346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cc 10426  ax-ac2 10454  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-acn 9933  df-ac 10107  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-ioo 13324  df-ico 13326  df-fl 13753  df-rest 17364  df-topgen 17385  df-top 22378  df-bases 22431  df-salg 44960  df-salgen 44964  df-smblfn 45347

This theorem is referenced by:  smfsuplem2  45463