Theorem ixpsnf1o 6533

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 4040	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V)
3		vex 2636	. . . . 5 ⊢ 𝑥 ∈ V
4	3	snex 4041	. . . 4 ⊢ {𝑥} ∈ V
5		xpexg 4581	. . . 4 ⊢ (({𝐼} ∈ V ∧ {𝑥} ∈ V) → ({𝐼} × {𝑥}) ∈ V)
6	2, 4, 5	sylancl 405	. . 3 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝑥}) ∈ V)
7	6	adantr 271	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ({𝐼} × {𝑥}) ∈ V)
8		vex 2636	. . . . 5 ⊢ 𝑎 ∈ V
9	8	rnex 4732	. . . 4 ⊢ ran 𝑎 ∈ V
10	9	uniex 4288	. . 3 ⊢ ∪ ran 𝑎 ∈ V
11	10	a1i 9	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴) → ∪ ran 𝑎 ∈ V)
12		sneq 3477	. . . . . 6 ⊢ (𝑏 = 𝐼 → {𝑏} = {𝐼})
13	12	xpeq1d 4490	. . . . 5 ⊢ (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
14	13	eqeq2d 2106	. . . 4 ⊢ (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
15	14	anbi2d 453	. . 3 ⊢ (𝑏 = 𝐼 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥}))))
16		elixpsn 6532	. . . . . 6 ⊢ (𝑏 ∈ V → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
17	16	elv 2637	. . . . 5 ⊢ (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
18	12	ixpeq1d 6507	. . . . . 6 ⊢ (𝑏 = 𝐼 → X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
19	18	eleq2d 2164	. . . . 5 ⊢ (𝑏 = 𝐼 → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
20	17, 19	syl5bbr 193	. . . 4 ⊢ (𝑏 = 𝐼 → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
21	20	anbi1d 454	. . 3 ⊢ (𝑏 = 𝐼 → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
22		vex 2636	. . . . . . 7 ⊢ 𝑏 ∈ V
23	22, 3	xpsn 5512	. . . . . 6 ⊢ ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
24	23	eqeq2i 2105	. . . . 5 ⊢ (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
25	24	anbi2i 446	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
26		eqid 2095	. . . . . . . . 9 ⊢ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
27		opeq2 3645	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
28	27	sneqd 3479	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
29	28	rspceeqv 2753	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
30	26, 29	mpan2 417	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
31	22, 3	op2nda 4949	. . . . . . . . 9 ⊢ ∪ ran {⟨𝑏, 𝑥⟩} = 𝑥
32	31	eqcomi 2099	. . . . . . . 8 ⊢ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}
33	30, 32	jctir 307	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
34		eqeq1 2101	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35	34	rexbidv 2392	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
36		rneq 4694	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
37	36	unieqd 3686	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑥⟩})
38	37	eqeq2d 2106	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
39	35, 38	anbi12d 458	. . . . . . 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 2636	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
43	22, 42	op2nda 4949	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑏, 𝑐⟩} = 𝑐
44	43	eqeq2i 2105	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
45		eqidd 2096	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
46	45	ancli 317	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐴 → (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
47		eleq1w 2155	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴))
48		opeq2 3645	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
49	48	sneqd 3479	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
50	49	eqeq2d 2106	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
51	47, 50	anbi12d 458	. . . . . . . . . 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 4694	. . . . . . . . . . 11 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
55	54	unieqd 3686	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑐⟩})
56	55	eqeq2d 2106	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑐⟩}))
57		eqeq1 2101	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
58	57	anbi2d 453	. . . . . . . . 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 2496	. . . . . 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 2694	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
66	1, 7, 11, 65	f1od 5885	1 ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∃wrex 2371  Vcvv 2633  {csn 3466  ⟨cop 3469  ∪ cuni 3675   ↦ cmpt 3921   × cxp 4465  ran crn 4468  –1-1-onto→wf1o 5048  Xcixp 6495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ixp 6496

This theorem is referenced by:  mapsnf1o  6534