Theorem uchoice 6230

Description: Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7326 (with the key difference being the change of ∃ to ∃!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)

Assertion

Ref	Expression
uchoice	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝜑,𝑓

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem uchoice

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2206	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	fnopabg 5405	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
3	2	biimpi 120	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
4	3	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
5		simpl 109	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → 𝐴 ∈ 𝑉)
6		fnex 5813	. . . . . 6 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
7	4, 5, 6	syl2anc 411	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
8		fnopfvb 5627	. . . . . . . . . 10 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}))
9		nfv 1552	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑢 ∈ 𝐴
10		nfsbc1v 3018	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[𝑢 / 𝑥]𝜑
11	9, 10	nfan 1589	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑢 ∈ 𝐴 ∧ [𝑢 / 𝑥]𝜑)
12		nfv 1552	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑢 ∈ 𝐴
13		nfsbc1v 3018	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦[𝑣 / 𝑦][𝑢 / 𝑥]𝜑
14	12, 13	nfan 1589	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)
15		vex 2776	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
16		vex 2776	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
17		eleq1w 2267	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
18		sbceq1a 3009	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝜑 ↔ [𝑢 / 𝑥]𝜑))
19	17, 18	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑢 ∈ 𝐴 ∧ [𝑢 / 𝑥]𝜑)))
20		sbceq1a 3009	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → ([𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
21	20	anbi2d 464	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → ((𝑢 ∈ 𝐴 ∧ [𝑢 / 𝑥]𝜑) ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))
22	11, 14, 15, 16, 19, 21	opelopabf 4325	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
23	8, 22	bitrdi 196	. . . . . . . . 9 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ 𝑢 ∈ 𝐴) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))
24	23	ralrimiva 2580	. . . . . . . 8 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 → ∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))
25	24	alrimiv 1898	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 → ∀𝑣∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))
26	25	ancli 323	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
27	4, 26	syl 14	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
28		fneq1 5367	. . . . . 6 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (𝑓 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴))
29		fveq1 5582	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (𝑓‘𝑢) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢))
30	29	eqeq1d 2215	. . . . . . . . 9 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ((𝑓‘𝑢) = 𝑣 ↔ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣))
31	30	bibi1d 233	. . . . . . . 8 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
32	31	ralbidv 2507	. . . . . . 7 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
33	32	albidv 1848	. . . . . 6 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ∀𝑣∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
34	28, 33	anbi12d 473	. . . . 5 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ((𝑓 Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))) ↔ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))))
35	7, 27, 34	elabd 2919	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
36		ralcom4 2795	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)))
37	36	anbi2i 457	. . . . 5 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
38	37	exbii 1629	. . . 4 ⊢ (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))) ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑣∀𝑢 ∈ 𝐴 ((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
39	35, 38	sylibr 134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))))
40		nfv 1552	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑢) = 𝑣
41		nfcv 2349	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
42	41, 10	nfsbc 3020	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑣 / 𝑦][𝑢 / 𝑥]𝜑
43	9, 42	nfan 1589	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)
44	40, 43	nfbi 1613	. . . . . . 7 ⊢ Ⅎ𝑥((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
45	44	nfal 1600	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
46		nfv 1552	. . . . . 6 ⊢ Ⅎ𝑢∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))
47		fveqeq2 5592	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑓‘𝑢) = 𝑣 ↔ (𝑓‘𝑥) = 𝑣))
48		eleq1w 2267	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
49		sbceq2a 3010	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → ([𝑢 / 𝑥]𝜑 ↔ 𝜑))
50	49	sbcbidv 3058	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → ([𝑣 / 𝑦][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑦]𝜑))
51	48, 50	anbi12d 473	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑) ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)))
52	47, 51	bibi12d 235	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))))
53	52	albidv 1848	. . . . . 6 ⊢ (𝑢 = 𝑥 → (∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))))
54	45, 46, 53	cbvral 2735	. . . . 5 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)))
55	54	anbi2i 457	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))))
56	55	exbii 1629	. . 3 ⊢ (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))) ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))))
57	39, 56	sylib 122	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))))
58		eqidd 2207	. . . . . . 7 ⊢ (∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → (𝑓‘𝑥) = (𝑓‘𝑥))
59		vex 2776	. . . . . . . . 9 ⊢ 𝑓 ∈ V
60		vex 2776	. . . . . . . . 9 ⊢ 𝑥 ∈ V
61	59, 60	fvex 5603	. . . . . . . 8 ⊢ (𝑓‘𝑥) ∈ V
62		eqeq2 2216	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑓‘𝑥) = 𝑣 ↔ (𝑓‘𝑥) = (𝑓‘𝑥)))
63		dfsbcq 3001	. . . . . . . . . 10 ⊢ (𝑣 = (𝑓‘𝑥) → ([𝑣 / 𝑦]𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
64	63	anbi2d 464	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑) ↔ (𝑥 ∈ 𝐴 ∧ [(𝑓‘𝑥) / 𝑦]𝜑)))
65	62, 64	bibi12d 235	. . . . . . . 8 ⊢ (𝑣 = (𝑓‘𝑥) → (((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) ↔ ((𝑓‘𝑥) = (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ [(𝑓‘𝑥) / 𝑦]𝜑))))
66	61, 65	spcv 2868	. . . . . . 7 ⊢ (∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → ((𝑓‘𝑥) = (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ [(𝑓‘𝑥) / 𝑦]𝜑)))
67	58, 66	mpbid 147	. . . . . 6 ⊢ (∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → (𝑥 ∈ 𝐴 ∧ [(𝑓‘𝑥) / 𝑦]𝜑))
68	67	simprd 114	. . . . 5 ⊢ (∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → [(𝑓‘𝑥) / 𝑦]𝜑)
69	68	ralimi 2570	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑)
70	69	anim2i 342	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))) → (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
71	70	eximi 1624	. 2 ⊢ (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))
72	57, 71	syl 14	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 [(𝑓‘𝑥) / 𝑦]𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  [wsbc 2999  ⟨cop 3637  {copab 4108   Fn wfn 5271  ‘cfv 5276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284

This theorem is referenced by: (None)