Theorem elixpsn 8725

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 4571	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑧} = {𝐴})
2	1	ixpeq1d 8697	. . 3 ⊢ (𝑧 = 𝐴 → X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
3	2	eleq2d 2824	. 2 ⊢ (𝑧 = 𝐴 → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝐴}𝐵))
4		opeq1 4804	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
5	4	sneqd 4573	. . . 4 ⊢ (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
6	5	eqeq2d 2749	. . 3 ⊢ (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
7	6	rexbidv 3226	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8		elex 3450	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 → 𝐹 ∈ V)
9		snex 5354	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
10		eleq1 2826	. . . . 5 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
11	9, 10	mpbiri 257	. . . 4 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
12	11	rexlimivw 3211	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
13		eleq1 2826	. . . 4 ⊢ (𝑤 = 𝐹 → (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝑧}𝐵))
14		eqeq1 2742	. . . . 5 ⊢ (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
15	14	rexbidv 3226	. . . 4 ⊢ (𝑤 = 𝐹 → (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
16		vex 3436	. . . . . 6 ⊢ 𝑤 ∈ V
17	16	elixp 8692	. . . . 5 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵))
18		vex 3436	. . . . . . 7 ⊢ 𝑧 ∈ V
19		fveq2 6774	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤‘𝑥) = (𝑤‘𝑧))
20	19	eleq1d 2823	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
21	18, 20	ralsn 4617	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
22	21	anbi2i 623	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
23		simpl 483	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
24		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤‘𝑦) = (𝑤‘𝑧))
25	24	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
26	18, 25	ralsn 4617	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
27	26	biimpri 227	. . . . . . . . . 10 ⊢ ((𝑤‘𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
28	27	adantl 482	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
29		ffnfv 6992	. . . . . . . . 9 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵))
30	23, 28, 29	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
31	18	fsn2 7008	. . . . . . . 8 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
32	30, 31	sylib 217	. . . . . . 7 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
33		opeq2 4805	. . . . . . . . 9 ⊢ (𝑦 = (𝑤‘𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤‘𝑧)⟩)
34	33	sneqd 4573	. . . . . . . 8 ⊢ (𝑦 = (𝑤‘𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤‘𝑧)⟩})
35	34	rspceeqv 3575	. . . . . . 7 ⊢ (((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
36	32, 35	syl 17	. . . . . 6 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
37		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
38	18, 37	fvsn 7053	. . . . . . . . . 10 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
39		id 22	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
40	38, 39	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
41	18, 37	fnsn 6492	. . . . . . . . 9 ⊢ {⟨𝑧, 𝑦⟩} Fn {𝑧}
42	40, 41	jctil 520	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
43		fneq1 6524	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
44		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
45	44	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤‘𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
46	43, 45	anbi12d 631	. . . . . . . 8 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
47	42, 46	syl5ibrcom 246	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵)))
48	47	rexlimiv 3209	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
49	36, 48	impbii 208	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
50	17, 22, 49	3bitri 297	. . . 4 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
51	13, 15, 50	vtoclbg 3507	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
52	8, 12, 51	pm5.21nii 380	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
53	3, 7, 52	vtoclbg 3507	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432  {csn 4561  ⟨cop 4567   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  Xcixp 8685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ixp 8686

This theorem is referenced by:  ixpsnf1o  8726  hoidmv1le  44132