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Theorem ixpsnf1o 8934
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 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
Assertion
Ref Expression
ixpsnf1o (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {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.
StepHypRef Expression
1 ixpsnf1o.f . 2 𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
2 snex 5431 . . . 4 {𝐼} ∈ V
3 snex 5431 . . . 4 {𝑥} ∈ V
42, 3xpex 7742 . . 3 ({𝐼} × {𝑥}) ∈ V
54a1i 11 . 2 ((𝐼𝑉𝑥𝐴) → ({𝐼} × {𝑥}) ∈ V)
6 vex 3478 . . . . 5 𝑎 ∈ V
76rnex 7905 . . . 4 ran 𝑎 ∈ V
87uniex 7733 . . 3 ran 𝑎 ∈ V
98a1i 11 . 2 ((𝐼𝑉𝑎X𝑦 ∈ {𝐼}𝐴) → ran 𝑎 ∈ V)
10 sneq 4638 . . . . . 6 (𝑏 = 𝐼 → {𝑏} = {𝐼})
1110xpeq1d 5705 . . . . 5 (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
1211eqeq2d 2743 . . . 4 (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
1312anbi2d 629 . . 3 (𝑏 = 𝐼 → ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = ({𝐼} × {𝑥}))))
14 elixpsn 8933 . . . . . 6 (𝑏 ∈ V → (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
1514elv 3480 . . . . 5 (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
1610ixpeq1d 8905 . . . . . 6 (𝑏 = 𝐼X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
1716eleq2d 2819 . . . . 5 (𝑏 = 𝐼 → (𝑎X𝑦 ∈ {𝑏}𝐴𝑎X𝑦 ∈ {𝐼}𝐴))
1815, 17bitr3id 284 . . . 4 (𝑏 = 𝐼 → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎X𝑦 ∈ {𝐼}𝐴))
1918anbi1d 630 . . 3 (𝑏 = 𝐼 → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
20 vex 3478 . . . . . . 7 𝑏 ∈ V
21 vex 3478 . . . . . . 7 𝑥 ∈ V
2220, 21xpsn 7141 . . . . . 6 ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
2322eqeq2i 2745 . . . . 5 (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
2423anbi2i 623 . . . 4 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
25 eqid 2732 . . . . . . . . 9 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
26 opeq2 4874 . . . . . . . . . . 11 (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
2726sneqd 4640 . . . . . . . . . 10 (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
2827rspceeqv 3633 . . . . . . . . 9 ((𝑥𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
2925, 28mpan2 689 . . . . . . . 8 (𝑥𝐴 → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3020, 21op2nda 6227 . . . . . . . . 9 ran {⟨𝑏, 𝑥⟩} = 𝑥
3130eqcomi 2741 . . . . . . . 8 𝑥 = ran {⟨𝑏, 𝑥⟩}
3229, 31jctir 521 . . . . . . 7 (𝑥𝐴 → (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩}))
33 eqeq1 2736 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
3433rexbidv 3178 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35 rneq 5935 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3635unieqd 4922 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3736eqeq2d 2743 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑥⟩}))
3834, 37anbi12d 631 . . . . . . 7 (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩})))
3932, 38syl5ibrcom 246 . . . . . 6 (𝑥𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎)))
4039imp 407 . . . . 5 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
41 vex 3478 . . . . . . . . . . 11 𝑐 ∈ V
4220, 41op2nda 6227 . . . . . . . . . 10 ran {⟨𝑏, 𝑐⟩} = 𝑐
4342eqeq2i 2745 . . . . . . . . 9 (𝑥 = ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
44 eqidd 2733 . . . . . . . . . . 11 (𝑐𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
4544ancli 549 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
46 eleq1w 2816 . . . . . . . . . . 11 (𝑥 = 𝑐 → (𝑥𝐴𝑐𝐴))
47 opeq2 4874 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
4847sneqd 4640 . . . . . . . . . . . 12 (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
4948eqeq2d 2743 . . . . . . . . . . 11 (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
5046, 49anbi12d 631 . . . . . . . . . 10 (𝑥 = 𝑐 → ((𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
5145, 50syl5ibrcom 246 . . . . . . . . 9 (𝑐𝐴 → (𝑥 = 𝑐 → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5243, 51biimtrid 241 . . . . . . . 8 (𝑐𝐴 → (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53 rneq 5935 . . . . . . . . . . 11 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5453unieqd 4922 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5554eqeq2d 2743 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑐⟩}))
56 eqeq1 2736 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
5756anbi2d 629 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5855, 57imbi12d 344 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
5952, 58syl5ibrcom 246 . . . . . . 7 (𝑐𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))))
6059rexlimiv 3148 . . . . . 6 (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})))
6160imp 407 . . . . 5 ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
6240, 61impbii 208 . . . 4 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6324, 62bitri 274 . . 3 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6413, 19, 63vtoclbg 3559 . 2 (𝐼𝑉 → ((𝑥𝐴𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
651, 5, 9, 64f1od 7660 1 (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  {csn 4628  cop 4634   cuni 4908  cmpt 5231   × cxp 5674  ran crn 5677  1-1-ontowf1o 6542  Xcixp 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ixp 8894
This theorem is referenced by:  mapsnf1o  8935
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