Theorem smfpimcclem 43438

Description: Lemma for smfpimcc 43439 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smfpimcclem.n	⊢ Ⅎ𝑛𝜑
smfpimcclem.z	⊢ 𝑍 ∈ 𝑉
smfpimcclem.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
smfpimcclem.c	⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦)
smfpimcclem.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))

Assertion

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

Distinct variable groups:   𝐴,ℎ   𝐴,𝑠,𝑦   𝐶,𝑠,𝑦   ℎ,𝐹   𝐹,𝑠,𝑦   ℎ,𝐻   𝑆,ℎ,𝑛   𝑆,𝑠,𝑦,𝑛   ℎ,𝑍,𝑛   𝑦,𝑍   𝜑,𝑦

Allowed substitution hints:   𝜑(ℎ,𝑛,𝑠)   𝐴(𝑛)   𝐶(ℎ,𝑛)   𝐹(𝑛)   𝐻(𝑦,𝑛,𝑠)   𝑉(𝑦,ℎ,𝑛,𝑠)   𝑊(𝑦,ℎ,𝑛,𝑠)   𝑍(𝑠)

Proof of Theorem smfpimcclem

Step	Hyp	Ref	Expression
1		smfpimcclem.n	. . 3 ⊢ Ⅎ𝑛𝜑
2		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑠𝑆
3	2	ssrab2f 41752	. . . 4 ⊢ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ⊆ 𝑆
4		eqid 2798	. . . . . . 7 ⊢ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}
5		smfpimcclem.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
6	4, 5	rabexd 5200	. . . . . 6 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ V)
7	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ V)
8		simpl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝜑)
9		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
10		eqid 2798	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) = (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})
11	10	elrnmpt1 5794	. . . . . . 7 ⊢ ((𝑛 ∈ 𝑍 ∧ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ V) → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
12	9, 7, 11	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
13	8, 12	jca 515	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝜑 ∧ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})))
14		eleq1 2877	. . . . . . . 8 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → (𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ↔ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})))
15	14	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) ↔ (𝜑 ∧ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))))
16		fveq2 6645	. . . . . . . 8 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → (𝐶‘𝑦) = (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
17		id 22	. . . . . . . 8 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → 𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})
18	16, 17	eleq12d 2884	. . . . . . 7 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → ((𝐶‘𝑦) ∈ 𝑦 ↔ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
19	15, 18	imbi12d 348	. . . . . 6 ⊢ (𝑦 = {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} → (((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦) ↔ ((𝜑 ∧ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})))
20		smfpimcclem.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦)
21	19, 20	vtoclg 3515	. . . . 5 ⊢ ({𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ V → ((𝜑 ∧ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
22	7, 13, 21	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})
23	3, 22	sseldi 3913	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ 𝑆)
24		smfpimcclem.h	. . 3 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
25	1, 23, 24	fmptdf 6858	. 2 ⊢ (𝜑 → 𝐻:𝑍⟶𝑆)
26		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑠𝐶
27		nfrab1 3337	. . . . . . . . 9 ⊢ Ⅎ𝑠{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}
28	26, 27	nffv 6655	. . . . . . . 8 ⊢ Ⅎ𝑠(𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})
29		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑠(◡(𝐹‘𝑛) “ 𝐴)
30		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎ𝑠dom (𝐹‘𝑛)
31	28, 30	nfin 4143	. . . . . . . . 9 ⊢ Ⅎ𝑠((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛))
32	29, 31	nfeq 2968	. . . . . . . 8 ⊢ Ⅎ𝑠(◡(𝐹‘𝑛) “ 𝐴) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛))
33		ineq1 4131	. . . . . . . . 9 ⊢ (𝑠 = (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) → (𝑠 ∩ dom (𝐹‘𝑛)) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛)))
34	33	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑠 = (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) → ((◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛)) ↔ (◡(𝐹‘𝑛) “ 𝐴) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛))))
35	28, 2, 32, 34	elrabf 3624	. . . . . . 7 ⊢ ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))} ↔ ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ 𝑆 ∧ (◡(𝐹‘𝑛) “ 𝐴) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛))))
36	22, 35	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ 𝑆 ∧ (◡(𝐹‘𝑛) “ 𝐴) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛))))
37	36	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ 𝐴) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛)))
38	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})))
39	22	elexd 3461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∈ V)
40	38, 39	fvmpt2d 6758	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
41	40	ineq1d 4138	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)) = ((𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}) ∩ dom (𝐹‘𝑛)))
42	37, 41	eqtr4d 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
43	42	ex 416	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))))
44	1, 43	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
45		smfpimcclem.z	. . . . . 6 ⊢ 𝑍 ∈ 𝑉
46	45	elexi 3460	. . . . 5 ⊢ 𝑍 ∈ V
47	46	mptex 6963	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) ∈ V
48	24, 47	eqeltri 2886	. . 3 ⊢ 𝐻 ∈ V
49		feq1 6468	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ:𝑍⟶𝑆 ↔ 𝐻:𝑍⟶𝑆))
50		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑛ℎ
51		nfmpt1 5128	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))}))
52	24, 51	nfcxfr 2953	. . . . . 6 ⊢ Ⅎ𝑛𝐻
53	50, 52	nfeq 2968	. . . . 5 ⊢ Ⅎ𝑛 ℎ = 𝐻
54		fveq1 6644	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ‘𝑛) = (𝐻‘𝑛))
55	54	ineq1d 4138	. . . . . 6 ⊢ (ℎ = 𝐻 → ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))
56	55	eqeq2d 2809	. . . . 5 ⊢ (ℎ = 𝐻 → ((◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) ↔ (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))))
57	53, 56	ralbid 3195	. . . 4 ⊢ (ℎ = 𝐻 → (∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)) ↔ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))))
58	49, 57	anbi12d 633	. . 3 ⊢ (ℎ = 𝐻 → ((ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))) ↔ (𝐻:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛)))))
59	48, 58	spcev 3555	. 2 ⊢ ((𝐻:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))) → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))
60	25, 44, 59	syl2anc 587	1 ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∩ cin 3880   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  ⟶wf 6320  ‘cfv 6324

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by:  smfpimcc  43439