Theorem ixpsnf1o 8959

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 5397	. . . 4 ⊢ {𝐼} ∈ V
3		snex 5397	. . . 4 ⊢ {𝑥} ∈ V
4	2, 3	xpex 7765	. . 3 ⊢ ({𝐼} × {𝑥}) ∈ V
5	4	a1i 11	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ({𝐼} × {𝑥}) ∈ V)
6		vex 3455	. . . . 5 ⊢ 𝑎 ∈ V
7	6	rnex 7920	. . . 4 ⊢ ran 𝑎 ∈ V
8	7	uniex 7756	. . 3 ⊢ ∪ ran 𝑎 ∈ V
9	8	a1i 11	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴) → ∪ ran 𝑎 ∈ V)
10		sneq 4594	. . . . . 6 ⊢ (𝑏 = 𝐼 → {𝑏} = {𝐼})
11	10	xpeq1d 5680	. . . . 5 ⊢ (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
12	11	eqeq2d 2772	. . . 4 ⊢ (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
13	12	anbi2d 642	. . 3 ⊢ (𝑏 = 𝐼 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥}))))
14		elixpsn 8958	. . . . . 6 ⊢ (𝑏 ∈ V → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
15	14	elv 3456	. . . . 5 ⊢ (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
16	10	ixpeq1d 8930	. . . . . 6 ⊢ (𝑏 = 𝐼 → X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
17	16	eleq2d 2847	. . . . 5 ⊢ (𝑏 = 𝐼 → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
18	15, 17	bitr3id 288	. . . 4 ⊢ (𝑏 = 𝐼 → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
19	18	anbi1d 643	. . 3 ⊢ (𝑏 = 𝐼 → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
20		vex 3455	. . . . . . 7 ⊢ 𝑏 ∈ V
21		vex 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
22	20, 21	xpsn 7139	. . . . . 6 ⊢ ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
23	22	eqeq2i 2774	. . . . 5 ⊢ (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
24	23	anbi2i 635	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
25		eqid 2761	. . . . . . . . 9 ⊢ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
26		opeq2 4834	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
27	26	sneqd 4596	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
28	27	rspceeqv 3599	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
29	25, 28	mpan2 704	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
30	20, 21	op2nda 6228	. . . . . . . . 9 ⊢ ∪ ran {⟨𝑏, 𝑥⟩} = 𝑥
31	30	eqcomi 2770	. . . . . . . 8 ⊢ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}
32	29, 31	jctir 530	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
33		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
34	33	rexbidv 3187	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35		rneq 5918	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
38	34, 37	anbi12d 644	. . . . . . 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 250	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎)))
40	39	imp 412	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
41		vex 3455	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
42	20, 41	op2nda 6228	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑏, 𝑐⟩} = 𝑐
43	42	eqeq2i 2774	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
44		eqidd 2762	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
45	44	ancli 558	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐴 → (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
46		eleq1w 2844	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴))
47		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
48	47	sneqd 4596	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
49	48	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
50	46, 49	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
51	45, 50	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
52	43, 51	biimtrid 245	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53		rneq 5918	. . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑐⟩}))
56		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
57	56	anbi2d 642	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
58	55, 57	imbi12d 347	. . . . . . . 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 250	. . . . . . 7 ⊢ (𝑐 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))))
60	59	rexlimiv 3157	. . . . . 6 ⊢ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})))
61	60	imp 412	. . . . 5 ⊢ ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
62	40, 61	impbii 212	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
63	24, 62	bitri 278	. . 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 7671	1 ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5649  ran crn 5652  –1-1-onto→wf1o 6536  Xcixp 8918

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 8919

This theorem is used by:  mapsnf1o  8960