Theorem ixpsnf1o 6710

Description: A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
ixpsnf1o.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))

Assertion

Ref	Expression
ixpsnf1o	⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

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

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem ixpsnf1o

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ixpsnf1o.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))
2		snexg 4168	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V)
3		vex 2733	. . . . 5 ⊢ 𝑥 ∈ V
4	3	snex 4169	. . . 4 ⊢ {𝑥} ∈ V
5		xpexg 4723	. . . 4 ⊢ (({𝐼} ∈ V ∧ {𝑥} ∈ V) → ({𝐼} × {𝑥}) ∈ V)
6	2, 4, 5	sylancl 411	. . 3 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝑥}) ∈ V)
7	6	adantr 274	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ({𝐼} × {𝑥}) ∈ V)
8		vex 2733	. . . . 5 ⊢ 𝑎 ∈ V
9	8	rnex 4876	. . . 4 ⊢ ran 𝑎 ∈ V
10	9	uniex 4420	. . 3 ⊢ ∪ ran 𝑎 ∈ V
11	10	a1i 9	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴) → ∪ ran 𝑎 ∈ V)
12		sneq 3592	. . . . . 6 ⊢ (𝑏 = 𝐼 → {𝑏} = {𝐼})
13	12	xpeq1d 4632	. . . . 5 ⊢ (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
14	13	eqeq2d 2182	. . . 4 ⊢ (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
15	14	anbi2d 461	. . 3 ⊢ (𝑏 = 𝐼 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥}))))
16		elixpsn 6709	. . . . . 6 ⊢ (𝑏 ∈ V → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
17	16	elv 2734	. . . . 5 ⊢ (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
18	12	ixpeq1d 6684	. . . . . 6 ⊢ (𝑏 = 𝐼 → X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
19	18	eleq2d 2240	. . . . 5 ⊢ (𝑏 = 𝐼 → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
20	17, 19	bitr3id 193	. . . 4 ⊢ (𝑏 = 𝐼 → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
21	20	anbi1d 462	. . 3 ⊢ (𝑏 = 𝐼 → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
22		vex 2733	. . . . . . 7 ⊢ 𝑏 ∈ V
23	22, 3	xpsn 5669	. . . . . 6 ⊢ ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
24	23	eqeq2i 2181	. . . . 5 ⊢ (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
25	24	anbi2i 454	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
26		eqid 2170	. . . . . . . . 9 ⊢ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
27		opeq2 3764	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
28	27	sneqd 3594	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
29	28	rspceeqv 2852	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
30	26, 29	mpan2 423	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
31	22, 3	op2nda 5093	. . . . . . . . 9 ⊢ ∪ ran {⟨𝑏, 𝑥⟩} = 𝑥
32	31	eqcomi 2174	. . . . . . . 8 ⊢ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}
33	30, 32	jctir 311	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
34		eqeq1 2177	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35	34	rexbidv 2471	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
36		rneq 4836	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
37	36	unieqd 3805	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑥⟩})
38	37	eqeq2d 2182	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
39	35, 38	anbi12d 470	. . . . . . 7 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩})))
40	33, 39	syl5ibrcom 156	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎)))
41	40	imp 123	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
42		vex 2733	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
43	22, 42	op2nda 5093	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑏, 𝑐⟩} = 𝑐
44	43	eqeq2i 2181	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
45		eqidd 2171	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
46	45	ancli 321	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐴 → (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
47		eleq1w 2231	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴))
48		opeq2 3764	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
49	48	sneqd 3594	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
50	49	eqeq2d 2182	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
51	47, 50	anbi12d 470	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
52	46, 51	syl5ibrcom 156	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53	44, 52	syl5bi 151	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
54		rneq 4836	. . . . . . . . . . 11 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
55	54	unieqd 3805	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑐⟩})
56	55	eqeq2d 2182	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑐⟩}))
57		eqeq1 2177	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
58	57	anbi2d 461	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
59	56, 58	imbi12d 233	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
60	53, 59	syl5ibrcom 156	. . . . . . 7 ⊢ (𝑐 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))))
61	60	rexlimiv 2581	. . . . . 6 ⊢ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})))
62	61	imp 123	. . . . 5 ⊢ ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
63	41, 62	impbii 125	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
64	25, 63	bitri 183	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
65	15, 21, 64	vtoclbg 2791	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
66	1, 7, 11, 65	f1od 6049	1 ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  ∃wrex 2449  Vcvv 2730  {csn 3581  ⟨cop 3584  ∪ cuni 3794   ↦ cmpt 4048   × cxp 4607  ran crn 4610  –1-1-onto→wf1o 5195  Xcixp 6672

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ixp 6673

This theorem is referenced by:  mapsnf1o  6711