Theorem elixpsn 8948

Description: Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
elixpsn	⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦

Proof of Theorem elixpsn

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4597	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑧} = {𝐴})
2	1	ixpeq1d 8920	. . 3 ⊢ (𝑧 = 𝐴 → X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
3	2	eleq2d 2848	. 2 ⊢ (𝑧 = 𝐴 → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝐴}𝐵))
4		opeq1 4836	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
5	4	sneqd 4599	. . . 4 ⊢ (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
6	5	eqeq2d 2773	. . 3 ⊢ (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
7	6	rexbidv 3188	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8		elex 3474	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 → 𝐹 ∈ V)
9		snex 5408	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
10		eleq1 2850	. . . . 5 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
11	9, 10	mpbiri 261	. . . 4 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
12	11	rexlimivw 3161	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13		eleq1 2850	. . . 4 ⊢ (𝑤 = 𝐹 → (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝑧}𝐵))
14		eqeq1 2766	. . . . 5 ⊢ (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
15	14	rexbidv 3188	. . . 4 ⊢ (𝑤 = 𝐹 → (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16		vex 3457	. . . . . 6 ⊢ 𝑤 ∈ V
17	16	elixp 8915	. . . . 5 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵))
18		vex 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
19		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤‘𝑥) = (𝑤‘𝑧))
20	19	eleq1d 2847	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
21	18, 20	ralsn 4645	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
22	21	anbi2i 635	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
23		simpl 488	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑤‘𝑦) = (𝑤‘𝑧))
25	24	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
26	18, 25	ralsn 4645	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
27	26	bilanri 512	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
28		ffnfv 7116	. . . . . . . . 9 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵))
29	23, 27, 28	sylanbrc 595	. . . . . . . 8 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
30	18	fsn2 7134	. . . . . . . 8 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
31	29, 30	sylib 221	. . . . . . 7 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
32		opeq2 4837	. . . . . . . . 9 ⊢ (𝑦 = (𝑤‘𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤‘𝑧)⟩)
33	32	sneqd 4599	. . . . . . . 8 ⊢ (𝑦 = (𝑤‘𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤‘𝑧)⟩})
34	33	rspceeqv 3602	. . . . . . 7 ⊢ (((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
35	31, 34	syl 18	. . . . . 6 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
36		vex 3457	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
37	18, 36	fvsn 7183	. . . . . . . . . 10 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
38		id 23	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
39	37, 38	eqeltrid 2866	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
40	18, 36	fnsn 6595	. . . . . . . . 9 ⊢ {⟨𝑧, 𝑦⟩} Fn {𝑧}
41	39, 40	jctil 529	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
42		fneq1 6627	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
43		fveq1 6881	. . . . . . . . . 10 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
44	43	eleq1d 2847	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤‘𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
45	42, 44	anbi12d 644	. . . . . . . 8 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
46	41, 45	syl5ibrcom 250	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵)))
47	46	rexlimiv 3158	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
48	35, 47	impbii 212	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
49	17, 22, 48	3bitri 300	. . . 4 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
50	13, 15, 49	vtoclbg 3522	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
51	8, 12, 50	pm5.21nii 381	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
52	3, 7, 51	vtoclbg 3522	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453  {csn 4587  ⟨cop 4593   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  Xcixp 8908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ixp 8909

This theorem is used by:  ixpsnf1o  8949  hoidmv1le  47430