Theorem ixpsnf1o 6984

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 4302	. . . 4 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V)
3		vex 2818	. . . . 5 ⊢ 𝑥 ∈ V
4	3	snex 4303	. . . 4 ⊢ {𝑥} ∈ V
5		xpexg 4869	. . . 4 ⊢ (({𝐼} ∈ V ∧ {𝑥} ∈ V) → ({𝐼} × {𝑥}) ∈ V)
6	2, 4, 5	sylancl 413	. . 3 ⊢ (𝐼 ∈ 𝑉 → ({𝐼} × {𝑥}) ∈ V)
7	6	adantr 276	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐴) → ({𝐼} × {𝑥}) ∈ V)
8		vex 2818	. . . . 5 ⊢ 𝑎 ∈ V
9	8	rnex 5030	. . . 4 ⊢ ran 𝑎 ∈ V
10	9	uniex 4563	. . 3 ⊢ ∪ ran 𝑎 ∈ V
11	10	a1i 9	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴) → ∪ ran 𝑎 ∈ V)
12		sneq 3705	. . . . . 6 ⊢ (𝑏 = 𝐼 → {𝑏} = {𝐼})
13	12	xpeq1d 4777	. . . . 5 ⊢ (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
14	13	eqeq2d 2246	. . . 4 ⊢ (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
15	14	anbi2d 464	. . 3 ⊢ (𝑏 = 𝐼 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥}))))
16		elixpsn 6983	. . . . . 6 ⊢ (𝑏 ∈ V → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
17	16	elv 2819	. . . . 5 ⊢ (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
18	12	ixpeq1d 6958	. . . . . 6 ⊢ (𝑏 = 𝐼 → X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
19	18	eleq2d 2304	. . . . 5 ⊢ (𝑏 = 𝐼 → (𝑎 ∈ X𝑦 ∈ {𝑏}𝐴 ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
20	17, 19	bitr3id 194	. . . 4 ⊢ (𝑏 = 𝐼 → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎 ∈ X𝑦 ∈ {𝐼}𝐴))
21	20	anbi1d 465	. . 3 ⊢ (𝑏 = 𝐼 → ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
22		vex 2818	. . . . . . 7 ⊢ 𝑏 ∈ V
23	22, 3	xpsn 5859	. . . . . 6 ⊢ ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
24	23	eqeq2i 2245	. . . . 5 ⊢ (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
25	24	anbi2i 457	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
26		eqid 2234	. . . . . . . . 9 ⊢ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
27		opeq2 3889	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
28	27	sneqd 3707	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
29	28	rspceeqv 2942	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
30	26, 29	mpan2 425	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
31	22, 3	op2nda 5252	. . . . . . . . 9 ⊢ ∪ ran {⟨𝑏, 𝑥⟩} = 𝑥
32	31	eqcomi 2238	. . . . . . . 8 ⊢ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}
33	30, 32	jctir 313	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
34		eqeq1 2241	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35	34	rexbidv 2545	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐 ∈ 𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
36		rneq 4989	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
37	36	unieqd 3930	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑥⟩})
38	37	eqeq2d 2246	. . . . . . . 8 ⊢ (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑥⟩}))
39	35, 38	anbi12d 473	. . . . . . 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 157	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎)))
41	40	imp 124	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
42		vex 2818	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
43	22, 42	op2nda 5252	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑏, 𝑐⟩} = 𝑐
44	43	eqeq2i 2245	. . . . . . . . 9 ⊢ (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
45		eqidd 2235	. . . . . . . . . . 11 ⊢ (𝑐 ∈ 𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
46	45	ancli 323	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐴 → (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
47		eleq1w 2295	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ↔ 𝑐 ∈ 𝐴))
48		opeq2 3889	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
49	48	sneqd 3707	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
50	49	eqeq2d 2246	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
51	47, 50	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑥 = 𝑐 → ((𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
52	46, 51	syl5ibrcom 157	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = 𝑐 → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53	44, 52	biimtrid 152	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → (𝑥 = ∪ ran {⟨𝑏, 𝑐⟩} → (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
54		rneq 4989	. . . . . . . . . . 11 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
55	54	unieqd 3930	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ∪ ran 𝑎 = ∪ ran {⟨𝑏, 𝑐⟩})
56	55	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 ↔ 𝑥 = ∪ ran {⟨𝑏, 𝑐⟩}))
57		eqeq1 2241	. . . . . . . . . 10 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
58	57	anbi2d 464	. . . . . . . . 9 ⊢ (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥 ∈ 𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
59	56, 58	imbi12d 234	. . . . . . . 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 157	. . . . . . 7 ⊢ (𝑐 ∈ 𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))))
61	60	rexlimiv 2656	. . . . . 6 ⊢ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ∪ ran 𝑎 → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩})))
62	61	imp 124	. . . . 5 ⊢ ((∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎) → (𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}))
63	41, 62	impbii 126	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
64	25, 63	bitri 184	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐 ∈ 𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ∪ ran 𝑎))
65	15, 21, 64	vtoclbg 2878	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎 ∈ X𝑦 ∈ {𝐼}𝐴 ∧ 𝑥 = ∪ ran 𝑎)))
66	1, 7, 11, 65	f1od 6266	1 ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∃wrex 2523  Vcvv 2815  {csn 3694  ⟨cop 3697  ∪ cuni 3919   ↦ cmpt 4176   × cxp 4752  ran crn 4755  –1-1-onto→wf1o 5356  Xcixp 6946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ixp 6947

This theorem is referenced by:  mapsnf1o  6985