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Theorem ixpsnf1o 8491
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 5323 . . . 4 {𝐼} ∈ V
3 snex 5323 . . . 4 {𝑥} ∈ V
42, 3xpex 7464 . . 3 ({𝐼} × {𝑥}) ∈ V
54a1i 11 . 2 ((𝐼𝑉𝑥𝐴) → ({𝐼} × {𝑥}) ∈ V)
6 vex 3498 . . . . 5 𝑎 ∈ V
76rnex 7605 . . . 4 ran 𝑎 ∈ V
87uniex 7454 . . 3 ran 𝑎 ∈ V
98a1i 11 . 2 ((𝐼𝑉𝑎X𝑦 ∈ {𝐼}𝐴) → ran 𝑎 ∈ V)
10 sneq 4569 . . . . . 6 (𝑏 = 𝐼 → {𝑏} = {𝐼})
1110xpeq1d 5578 . . . . 5 (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
1211eqeq2d 2832 . . . 4 (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
1312anbi2d 628 . . 3 (𝑏 = 𝐼 → ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = ({𝐼} × {𝑥}))))
14 elixpsn 8490 . . . . . 6 (𝑏 ∈ V → (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
1514elv 3500 . . . . 5 (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
1610ixpeq1d 8462 . . . . . 6 (𝑏 = 𝐼X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
1716eleq2d 2898 . . . . 5 (𝑏 = 𝐼 → (𝑎X𝑦 ∈ {𝑏}𝐴𝑎X𝑦 ∈ {𝐼}𝐴))
1815, 17syl5bbr 286 . . . 4 (𝑏 = 𝐼 → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎X𝑦 ∈ {𝐼}𝐴))
1918anbi1d 629 . . 3 (𝑏 = 𝐼 → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
20 vex 3498 . . . . . . 7 𝑏 ∈ V
21 vex 3498 . . . . . . 7 𝑥 ∈ V
2220, 21xpsn 6896 . . . . . 6 ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
2322eqeq2i 2834 . . . . 5 (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
2423anbi2i 622 . . . 4 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
25 eqid 2821 . . . . . . . . 9 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
26 opeq2 4798 . . . . . . . . . . 11 (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
2726sneqd 4571 . . . . . . . . . 10 (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
2827rspceeqv 3637 . . . . . . . . 9 ((𝑥𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
2925, 28mpan2 687 . . . . . . . 8 (𝑥𝐴 → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3020, 21op2nda 6079 . . . . . . . . 9 ran {⟨𝑏, 𝑥⟩} = 𝑥
3130eqcomi 2830 . . . . . . . 8 𝑥 = ran {⟨𝑏, 𝑥⟩}
3229, 31jctir 521 . . . . . . 7 (𝑥𝐴 → (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩}))
33 eqeq1 2825 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
3433rexbidv 3297 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
35 rneq 5800 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3635unieqd 4842 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3736eqeq2d 2832 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑥⟩}))
3834, 37anbi12d 630 . . . . . . 7 (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩})))
3932, 38syl5ibrcom 248 . . . . . 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 3498 . . . . . . . . . . 11 𝑐 ∈ V
4220, 41op2nda 6079 . . . . . . . . . 10 ran {⟨𝑏, 𝑐⟩} = 𝑐
4342eqeq2i 2834 . . . . . . . . 9 (𝑥 = ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
44 eqidd 2822 . . . . . . . . . . 11 (𝑐𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
4544ancli 549 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
46 eleq1w 2895 . . . . . . . . . . 11 (𝑥 = 𝑐 → (𝑥𝐴𝑐𝐴))
47 opeq2 4798 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
4847sneqd 4571 . . . . . . . . . . . 12 (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
4948eqeq2d 2832 . . . . . . . . . . 11 (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
5046, 49anbi12d 630 . . . . . . . . . 10 (𝑥 = 𝑐 → ((𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
5145, 50syl5ibrcom 248 . . . . . . . . 9 (𝑐𝐴 → (𝑥 = 𝑐 → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5243, 51syl5bi 243 . . . . . . . 8 (𝑐𝐴 → (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
53 rneq 5800 . . . . . . . . . . 11 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5453unieqd 4842 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5554eqeq2d 2832 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑐⟩}))
56 eqeq1 2825 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
5756anbi2d 628 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5855, 57imbi12d 346 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
5952, 58syl5ibrcom 248 . . . . . . 7 (𝑐𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))))
6059rexlimiv 3280 . . . . . 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 210 . . . 4 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6324, 62bitri 276 . . 3 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6413, 19, 63vtoclbg 3569 . 2 (𝐼𝑉 → ((𝑥𝐴𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
651, 5, 9, 64f1od 7386 1 (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3139  Vcvv 3495  {csn 4559  cop 4565   cuni 4832  cmpt 5138   × cxp 5547  ran crn 5550  1-1-ontowf1o 6348  Xcixp 8450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ixp 8451
This theorem is referenced by:  mapsnf1o  8492
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