Theorem elixpsn 6637

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 3543	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑧} = {𝐴})
2	1	ixpeq1d 6612	. . 3 ⊢ (𝑧 = 𝐴 → X𝑥 ∈ {𝑧}𝐵 = X𝑥 ∈ {𝐴}𝐵)
3	2	eleq2d 2210	. 2 ⊢ (𝑧 = 𝐴 → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝐴}𝐵))
4		opeq1 3713	. . . . 5 ⊢ (𝑧 = 𝐴 → ⟨𝑧, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
5	4	sneqd 3545	. . . 4 ⊢ (𝑧 = 𝐴 → {⟨𝑧, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
6	5	eqeq2d 2152	. . 3 ⊢ (𝑧 = 𝐴 → (𝐹 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, 𝑦⟩}))
7	6	rexbidv 2439	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))
8		elex 2700	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 → 𝐹 ∈ V)
9		vex 2692	. . . . . . 7 ⊢ 𝑧 ∈ V
10		vex 2692	. . . . . . 7 ⊢ 𝑦 ∈ V
11	9, 10	opex 4159	. . . . . 6 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
12	11	snex 4117	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
13		eleq1 2203	. . . . 5 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → (𝐹 ∈ V ↔ {⟨𝑧, 𝑦⟩} ∈ V))
14	12, 13	mpbiri 167	. . . 4 ⊢ (𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
15	14	rexlimivw 2548	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩} → 𝐹 ∈ V)
16		eleq1 2203	. . . 4 ⊢ (𝑤 = 𝐹 → (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ 𝐹 ∈ X𝑥 ∈ {𝑧}𝐵))
17		eqeq1 2147	. . . . 5 ⊢ (𝑤 = 𝐹 → (𝑤 = {⟨𝑧, 𝑦⟩} ↔ 𝐹 = {⟨𝑧, 𝑦⟩}))
18	17	rexbidv 2439	. . . 4 ⊢ (𝑤 = 𝐹 → (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
19		vex 2692	. . . . . 6 ⊢ 𝑤 ∈ V
20	19	elixp 6607	. . . . 5 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵))
21		fveq2 5429	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑤‘𝑥) = (𝑤‘𝑧))
22	21	eleq1d 2209	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
23	9, 22	ralsn 3574	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
24	23	anbi2i 453	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ ∀𝑥 ∈ {𝑧} (𝑤‘𝑥) ∈ 𝐵) ↔ (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
25		simpl 108	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤 Fn {𝑧})
26		fveq2 5429	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤‘𝑦) = (𝑤‘𝑧))
27	26	eleq1d 2209	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵))
28	9, 27	ralsn 3574	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵 ↔ (𝑤‘𝑧) ∈ 𝐵)
29	28	biimpri 132	. . . . . . . . . 10 ⊢ ((𝑤‘𝑧) ∈ 𝐵 → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
30	29	adantl 275	. . . . . . . . 9 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵)
31		ffnfv 5586	. . . . . . . . 9 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ (𝑤 Fn {𝑧} ∧ ∀𝑦 ∈ {𝑧} (𝑤‘𝑦) ∈ 𝐵))
32	25, 30, 31	sylanbrc 414	. . . . . . . 8 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → 𝑤:{𝑧}⟶𝐵)
33	9	fsn2 5602	. . . . . . . 8 ⊢ (𝑤:{𝑧}⟶𝐵 ↔ ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
34	32, 33	sylib 121	. . . . . . 7 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}))
35		opeq2 3714	. . . . . . . . 9 ⊢ (𝑦 = (𝑤‘𝑧) → ⟨𝑧, 𝑦⟩ = ⟨𝑧, (𝑤‘𝑧)⟩)
36	35	sneqd 3545	. . . . . . . 8 ⊢ (𝑦 = (𝑤‘𝑧) → {⟨𝑧, 𝑦⟩} = {⟨𝑧, (𝑤‘𝑧)⟩})
37	36	rspceeqv 2811	. . . . . . 7 ⊢ (((𝑤‘𝑧) ∈ 𝐵 ∧ 𝑤 = {⟨𝑧, (𝑤‘𝑧)⟩}) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
38	34, 37	syl 14	. . . . . 6 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
39	9, 10	fvsn 5623	. . . . . . . . . 10 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
40		id 19	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
41	39, 40	eqeltrid 2227	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)
42	9, 10	fnsn 5185	. . . . . . . . 9 ⊢ {⟨𝑧, 𝑦⟩} Fn {𝑧}
43	41, 42	jctil 310	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
44		fneq1 5219	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ↔ {⟨𝑧, 𝑦⟩} Fn {𝑧}))
45		fveq1 5428	. . . . . . . . . 10 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
46	45	eleq1d 2209	. . . . . . . . 9 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤‘𝑧) ∈ 𝐵 ↔ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵))
47	44, 46	anbi12d 465	. . . . . . . 8 ⊢ (𝑤 = {⟨𝑧, 𝑦⟩} → ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ({⟨𝑧, 𝑦⟩} Fn {𝑧} ∧ ({⟨𝑧, 𝑦⟩}‘𝑧) ∈ 𝐵)))
48	43, 47	syl5ibrcom 156	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵)))
49	48	rexlimiv 2546	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩} → (𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵))
50	38, 49	impbii 125	. . . . 5 ⊢ ((𝑤 Fn {𝑧} ∧ (𝑤‘𝑧) ∈ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
51	20, 24, 50	3bitri 205	. . . 4 ⊢ (𝑤 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑤 = {⟨𝑧, 𝑦⟩})
52	16, 18, 51	vtoclbg 2750	. . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩}))
53	8, 15, 52	pm5.21nii 694	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ {𝑧}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝑧, 𝑦⟩})
54	3, 7, 53	vtoclbg 2750	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {⟨𝐴, 𝑦⟩}))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689  {csn 3532  ⟨cop 3535   Fn wfn 5126  ⟶wf 5127  ‘cfv 5131  Xcixp 6600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ixp 6601

This theorem is referenced by:  ixpsnf1o  6638