Theorem smfpimcc 46764

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 45003	. . . . . 6 ⊢ 𝑍 ≼ ω
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
4		mptct 10576	. . . . 5 ⊢ (𝑍 ≼ ω → (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
5		rnct 10563	. . . . 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 3482	. . . . . . . 8 ⊢ 𝑦 ∈ V
8		eqid 2735	. . . . . . . . 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 5972	. . . . . . . 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 1137	. . . . . . . . 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 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
18		eqid 2735	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑚) “ 𝐴)
24	15, 17, 18, 19, 20, 22, 23	smfpimbor1 46756	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
25		fvex 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑚) ∈ V
26	25	dmex 7932	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑚) ∈ V
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
28		elrest 17474	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 4395	. . . . . . . . . . 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 1131	. . . . . . . . 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 3006	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 ≠ ∅)
36	35	3exp 1118	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑍 → (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅)))
37	36	rexlimdv 3151	. . . . . 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 45173	. . 3 ⊢ (𝜑 → ∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
41		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
42		nfmpt1 5256	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
43	42	nfrn 5966	. . . . . . . 8 ⊢ Ⅎ𝑚ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
44		nfv 1912	. . . . . . . 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 1897	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
47	1	fvexi 6921	. . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
50		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
51	49, 50	eleq12d 2833	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑤) ∈ 𝑤))
52	51	rspccva 3621	. . . . . . 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 2735	. . . . . 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 46763	. . . . 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 1931	. . 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 2903	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑚
61	59, 60	nffv 6917	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐹‘𝑚)
62	61	nfcnv 5892	. . . . . . 7 ⊢ Ⅎ𝑛◡(𝐹‘𝑚)
63		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
64	62, 63	nfima 6088	. . . . . 6 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴)
65		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑛(ℎ‘𝑚)
66	61	nfdm 5965	. . . . . . 7 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
67	65, 66	nfin 4232	. . . . . 6 ⊢ Ⅎ𝑛((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
68	64, 67	nfeq 2917	. . . . 5 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
69		nfv 1912	. . . . 5 ⊢ Ⅎ𝑚(◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))
70		fveq2 6907	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
71	70	cnveqd 5889	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ◡(𝐹‘𝑚) = ◡(𝐹‘𝑛))
72	71	imaeq1d 6079	. . . . . 6 ⊢ (𝑚 = 𝑛 → (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑛) “ 𝐴))
73		fveq2 6907	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (ℎ‘𝑚) = (ℎ‘𝑛))
74	70	dmeqd 5919	. . . . . . 7 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
75	73, 74	ineq12d 4229	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚)) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))
76	72, 75	eqeq12d 2751	. . . . 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 623	. . 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 1845	. 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 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Ⅎwnfc 2888   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ∩ cin 3962  ∅c0 4339   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ωcom 7887   ≼ cdom 8982  ℤ_≥cuz 12876  (,)cioo 13384   ↾_t crest 17467  topGenctg 17484  SAlgcsalg 46264  SalGencsalgen 46268  SMblFncsmblfn 46651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-ioo 13388  df-ico 13390  df-fl 13829  df-rest 17469  df-topgen 17490  df-top 22916  df-bases 22969  df-salg 46265  df-salgen 46269  df-smblfn 46652

This theorem is referenced by:  smfsuplem2  46768