Theorem smfpimcc 46823

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 45068	. . . . . 6 ⊢ 𝑍 ≼ ω
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
4		mptct 10578	. . . . 5 ⊢ (𝑍 ≼ ω → (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))}) ≼ ω)
5		rnct 10565	. . . . 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 3484	. . . . . . . 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 5969	. . . . . . . 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 7104	. . . . . . . . . . . . 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 46815	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (◡(𝐹‘𝑚) “ 𝐴) ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
25		fvex 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑚) ∈ V
26	25	dmex 7931	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑚) ∈ V
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
28		elrest 17472	. . . . . . . . . . . . . 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 4389	. . . . . . . . . . 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 3008	. . . . . . . 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 3153	. . . . . 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 45235	. . 3 ⊢ (𝜑 → ∃𝑓∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
41		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
42		nfmpt1 5250	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
43	42	nfrn 5963	. . . . . . . 8 ⊢ Ⅎ𝑚ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})
44		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑓‘𝑦) ∈ 𝑦
45	43, 44	nfralw 3311	. . . . . . 7 ⊢ Ⅎ𝑚∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦
46	41, 45	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑚))})(𝑓‘𝑦) ∈ 𝑦)
47	1	fvexi 6920	. . . . . 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 6906	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑓‘𝑦) = (𝑓‘𝑤))
50		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
51	49, 50	eleq12d 2835	. . . . . . . 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 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 46822	. . . . 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 1933	. . 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 2905	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑚
61	59, 60	nffv 6916	. . . . . . . 8 ⊢ Ⅎ𝑛(𝐹‘𝑚)
62	61	nfcnv 5889	. . . . . . 7 ⊢ Ⅎ𝑛◡(𝐹‘𝑚)
63		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑛𝐴
64	62, 63	nfima 6086	. . . . . 6 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴)
65		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑛(ℎ‘𝑚)
66	61	nfdm 5962	. . . . . . 7 ⊢ Ⅎ𝑛dom (𝐹‘𝑚)
67	65, 66	nfin 4224	. . . . . 6 ⊢ Ⅎ𝑛((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
68	64, 67	nfeq 2919	. . . . 5 ⊢ Ⅎ𝑛(◡(𝐹‘𝑚) “ 𝐴) = ((ℎ‘𝑚) ∩ dom (𝐹‘𝑚))
69		nfv 1914	. . . . 5 ⊢ Ⅎ𝑚(◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))
70		fveq2 6906	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
71	70	cnveqd 5886	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ◡(𝐹‘𝑚) = ◡(𝐹‘𝑛))
72	71	imaeq1d 6077	. . . . . 6 ⊢ (𝑚 = 𝑛 → (◡(𝐹‘𝑚) “ 𝐴) = (◡(𝐹‘𝑛) “ 𝐴))
73		fveq2 6906	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (ℎ‘𝑚) = (ℎ‘𝑛))
74	70	dmeqd 5916	. . . . . . 7 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
75	73, 74	ineq12d 4221	. . . . . 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 3306	. . . 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 1848	. 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 1540  ∃wex 1779   ∈ wcel 2108  Ⅎwnfc 2890   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∩ cin 3950  ∅c0 4333   class class class wbr 5143   ↦ cmpt 5225  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ωcom 7887   ≼ cdom 8983  ℤ_≥cuz 12878  (,)cioo 13387   ↾_t crest 17465  topGenctg 17482  SAlgcsalg 46323  SalGencsalgen 46327  SMblFncsmblfn 46710

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-fl 13832  df-rest 17467  df-topgen 17488  df-top 22900  df-bases 22953  df-salg 46324  df-salgen 46328  df-smblfn 46711

This theorem is referenced by:  smfsuplem2  46827