Theorem ixpsnf1o 8888

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		snex 5386	. . . 4 ⊢ {𝐼} ∈ V
3		snex 5386	. . . 4 ⊢ {𝑥} ∈ V
4	2, 3	xpex 7709	. . 3 ⊢ ({𝐼} × {𝑥}) ∈ V
5	4	a1i 11	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ({𝐼} × {𝑥}) ∈ V)
6		vex 3448	. . . . 5 ⊢ 𝑎 ∈ V
7	6	rnex 7866	. . . 4 ⊢ ran 𝑎 ∈ V
8	7	uniex 7697	. . 3 ⊢ ∪ ran 𝑎 ∈ V
9	8	a1i 11	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴) → ∪ ran 𝑎 ∈ V)
10		sneq 4595	. . . . . 6 ⊢ (𝑏 = 𝐼 → {𝑏} = {𝐼})
11	10	xpeq1d 5660	. . . . 5 ⊢ (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
12	11	eqeq2d 2740	. . . 4 ⊢ (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
13	12	anbi2d 630	. . 3 ⊢ (𝑏 = 𝐼 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥}))))
14		elixpsn 8887	. . . . . 6 ⊢ (𝑏 ∈ V → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
15	14	elv 3449	. . . . 5 ⊢ (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
16	10	ixpeq1d 8859	. . . . . 6 ⊢ (𝑏 = 𝐼 → X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
17	16	eleq2d 2814	. . . . 5 ⊢ (𝑏 = 𝐼 → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
18	15, 17	bitr3id 285	. . . 4 ⊢ (𝑏 = 𝐼 → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
19	18	anbi1d 631	. . 3 ⊢ (𝑏 = 𝐼 → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
20		vex 3448	. . . . . . 7 ⊢ 𝑏 ∈ V
21		vex 3448	. . . . . . 7 ⊢ 𝑥 ∈ V
22	20, 21	xpsn 7095	. . . . . 6 ⊢ ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
23	22	eqeq2i 2742	. . . . 5 ⊢ (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
24	23	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
25		eqid 2729	. . . . . . . . 9 ⊢ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
26		opeq2 4834	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
27	26	sneqd 4597	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
28	27	rspceeqv 3608	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
29	25, 28	mpan2 691	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
30	20, 21	op2nda 6189	. . . . . . . . 9 ⊢ ∪ ran {⟨𝑏, 𝑥⟩} = 𝑥
31	30	eqcomi 2738	. . . . . . . 8 ⊢ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}
32	29, 31	jctir 520	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
33		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
34	33	rexbidv 3157	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35		rneq 5889	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
36	35	unieqd 4880	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑥⟩})
37	36	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
38	34, 37	anbi12d 632	. . . . . . 7 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩})))
39	32, 38	syl5ibrcom 247	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎)))
40	39	imp 406	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
41		vex 3448	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
42	20, 41	op2nda 6189	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑏, 𝑐⟩} = 𝑐
43	42	eqeq2i 2742	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
44		eqidd 2730	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
45	44	ancli 548	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐴 → (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
46		eleq1w 2811	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴))
47		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
48	47	sneqd 4597	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
49	48	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
50	46, 49	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
51	45, 50	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
52	43, 51	biimtrid 242	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53		rneq 5889	. . . . . . . . . . 11 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
54	53	unieqd 4880	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑐⟩})
55	54	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑐⟩}))
56		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
57	56	anbi2d 630	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
58	55, 57	imbi12d 344	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
59	52, 58	syl5ibrcom 247	. . . . . . 7 ⊢ (𝑐 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))))
60	59	rexlimiv 3127	. . . . . 6 ⊢ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})))
61	60	imp 406	. . . . 5 ⊢ ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
62	40, 61	impbii 209	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
63	24, 62	bitri 275	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
64	13, 19, 63	vtoclbg 3520	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
65	1, 5, 9, 64	f1od 7621	1 ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444  {csn 4585  ⟨cop 4591  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5629  ran crn 5632  –1-1-onto→wf1o 6498  Xcixp 8847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ixp 8848

This theorem is referenced by:  mapsnf1o  8889