Theorem uchoice 6195

Description: Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7274 (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 2196	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2	1	fnopabg 5381	. . . . . . . 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 5784	. . . . . 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 5602	. . . . . . . . . 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 1542	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑢 ∈ 𝐴
10		nfsbc1v 3008	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥[𝑢 / 𝑥]𝜑
11	9, 10	nfan 1579	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑢 ∈ 𝐴 ∧ [𝑢 / 𝑥]𝜑)
12		nfv 1542	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑢 ∈ 𝐴
13		nfsbc1v 3008	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦[𝑣 / 𝑦][𝑢 / 𝑥]𝜑
14	12, 13	nfan 1579	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)
15		vex 2766	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
16		vex 2766	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
17		eleq1w 2257	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴))
18		sbceq1a 2999	. . . . . . . . . . . 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 2999	. . . . . . . . . . . 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 4309	. . . . . . . . . 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 2570	. . . . . . . 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 1888	. . . . . . 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 5346	. . . . . 6 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (𝑓 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴))
29		fveq1 5557	. . . . . . . . . 10 ⊢ (𝑓 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → (𝑓‘𝑢) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}‘𝑢))
30	29	eqeq1d 2205	. . . . . . . . 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 2497	. . . . . . 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 1838	. . . . . 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 2909	. . . 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 2785	. . . . . 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 1619	. . . 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 1542	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑢) = 𝑣
41		nfcv 2339	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑣
42	41, 10	nfsbc 3010	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑣 / 𝑦][𝑢 / 𝑥]𝜑
43	9, 42	nfan 1579	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑)
44	40, 43	nfbi 1603	. . . . . . 7 ⊢ Ⅎ𝑥((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
45	44	nfal 1590	. . . . . 6 ⊢ Ⅎ𝑥∀𝑣((𝑓‘𝑢) = 𝑣 ↔ (𝑢 ∈ 𝐴 ∧ [𝑣 / 𝑦][𝑢 / 𝑥]𝜑))
46		nfv 1542	. . . . . 6 ⊢ Ⅎ𝑢∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑))
47		fveqeq2 5567	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑓‘𝑢) = 𝑣 ↔ (𝑓‘𝑥) = 𝑣))
48		eleq1w 2257	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
49		sbceq2a 3000	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → ([𝑢 / 𝑥]𝜑 ↔ 𝜑))
50	49	sbcbidv 3048	. . . . . . . . 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 1838	. . . . . 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 2725	. . . . 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 1619	. . 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 2197	. . . . . . 7 ⊢ (∀𝑣((𝑓‘𝑥) = 𝑣 ↔ (𝑥 ∈ 𝐴 ∧ [𝑣 / 𝑦]𝜑)) → (𝑓‘𝑥) = (𝑓‘𝑥))
59		vex 2766	. . . . . . . . 9 ⊢ 𝑓 ∈ V
60		vex 2766	. . . . . . . . 9 ⊢ 𝑥 ∈ V
61	59, 60	fvex 5578	. . . . . . . 8 ⊢ (𝑓‘𝑥) ∈ V
62		eqeq2 2206	. . . . . . . . 9 ⊢ (𝑣 = (𝑓‘𝑥) → ((𝑓‘𝑥) = 𝑣 ↔ (𝑓‘𝑥) = (𝑓‘𝑥)))
63		dfsbcq 2991	. . . . . . . . . 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 2858	. . . . . . 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 2560	. . . 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 1614	. 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 1362   = wceq 1364  ∃wex 1506  ∃!weu 2045   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  [wsbc 2989  ⟨cop 3625  {copab 4093   Fn wfn 5253  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266

This theorem is referenced by: (None)