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Theorem f1ofvswap 7315
Description: Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
Assertion
Ref Expression
f1ofvswap ((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}):𝐴1-1-onto𝐵)

Proof of Theorem f1ofvswap
StepHypRef Expression
1 f1oi 6877 . . . . . 6 ( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌})
2 f1oprswap 6883 . . . . . 6 ((𝑋𝐴𝑌𝐴) → {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌})
3 disjdifr 4473 . . . . . . 7 ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅
4 f1oun 6858 . . . . . . 7 (((( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌}) ∧ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌}) ∧ (((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅ ∧ ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅)) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
53, 3, 4mpanr12 704 . . . . . 6 ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌}) ∧ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌}) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
61, 2, 5sylancr 586 . . . . 5 ((𝑋𝐴𝑌𝐴) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
7 prssi 4825 . . . . . . 7 ((𝑋𝐴𝑌𝐴) → {𝑋, 𝑌} ⊆ 𝐴)
8 undifr 4483 . . . . . . 7 ({𝑋, 𝑌} ⊆ 𝐴 ↔ ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
97, 8sylib 217 . . . . . 6 ((𝑋𝐴𝑌𝐴) → ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
10 f1oeq23 6830 . . . . . 6 ((((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴 ∧ ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴) → ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) ↔ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴1-1-onto𝐴))
119, 9, 10syl2anc 583 . . . . 5 ((𝑋𝐴𝑌𝐴) → ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) ↔ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴1-1-onto𝐴))
126, 11mpbid 231 . . . 4 ((𝑋𝐴𝑌𝐴) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴1-1-onto𝐴)
13 f1oco 6862 . . . 4 ((𝐹:𝐴1-1-onto𝐵 ∧ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴1-1-onto𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴1-1-onto𝐵)
1412, 13sylan2 592 . . 3 ((𝐹:𝐴1-1-onto𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴1-1-onto𝐵)
15143impb 1113 . 2 ((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴1-1-onto𝐵)
16 f1ofn 6840 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)
17 coundi 6251 . . . . . 6 (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}))
18 fcoconst 7143 . . . . . . . . . . 11 ((𝐹 Fn 𝐴𝑌𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹𝑌)}))
19183adant2 1129 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹𝑌)}))
20 xpsng 7148 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐴) → ({𝑋} × {𝑌}) = {⟨𝑋, 𝑌⟩})
2120coeq2d 5865 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
22213adant1 1128 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
23 fvex 6910 . . . . . . . . . . . 12 (𝐹𝑌) ∈ V
24 xpsng 7148 . . . . . . . . . . . 12 ((𝑋𝐴 ∧ (𝐹𝑌) ∈ V) → ({𝑋} × {(𝐹𝑌)}) = {⟨𝑋, (𝐹𝑌)⟩})
2523, 24mpan2 690 . . . . . . . . . . 11 (𝑋𝐴 → ({𝑋} × {(𝐹𝑌)}) = {⟨𝑋, (𝐹𝑌)⟩})
26253ad2ant2 1132 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ({𝑋} × {(𝐹𝑌)}) = {⟨𝑋, (𝐹𝑌)⟩})
2719, 22, 263eqtr3d 2776 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩}) = {⟨𝑋, (𝐹𝑌)⟩})
28 fcoconst 7143 . . . . . . . . . . 11 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹𝑋)}))
29283adant3 1130 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹𝑋)}))
30 xpsng 7148 . . . . . . . . . . . . 13 ((𝑌𝐴𝑋𝐴) → ({𝑌} × {𝑋}) = {⟨𝑌, 𝑋⟩})
3130coeq2d 5865 . . . . . . . . . . . 12 ((𝑌𝐴𝑋𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
3231ancoms 458 . . . . . . . . . . 11 ((𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
33323adant1 1128 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
34 fvex 6910 . . . . . . . . . . . 12 (𝐹𝑋) ∈ V
35 xpsng 7148 . . . . . . . . . . . 12 ((𝑌𝐴 ∧ (𝐹𝑋) ∈ V) → ({𝑌} × {(𝐹𝑋)}) = {⟨𝑌, (𝐹𝑋)⟩})
3634, 35mpan2 690 . . . . . . . . . . 11 (𝑌𝐴 → ({𝑌} × {(𝐹𝑋)}) = {⟨𝑌, (𝐹𝑋)⟩})
37363ad2ant3 1133 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ({𝑌} × {(𝐹𝑋)}) = {⟨𝑌, (𝐹𝑋)⟩})
3829, 33, 373eqtr3d 2776 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ {⟨𝑌, 𝑋⟩}) = {⟨𝑌, (𝐹𝑋)⟩})
3927, 38uneq12d 4163 . . . . . . . 8 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩})) = ({⟨𝑋, (𝐹𝑌)⟩} ∪ {⟨𝑌, (𝐹𝑋)⟩}))
40 df-pr 4632 . . . . . . . . . 10 {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩} = ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})
4140coeq2i 5863 . . . . . . . . 9 (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩}))
42 coundi 6251 . . . . . . . . 9 (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
4341, 42eqtri 2756 . . . . . . . 8 (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
44 df-pr 4632 . . . . . . . 8 {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩} = ({⟨𝑋, (𝐹𝑌)⟩} ∪ {⟨𝑌, (𝐹𝑋)⟩})
4539, 43, 443eqtr4g 2793 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩})
4645uneq2d 4162 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}))
4717, 46eqtrid 2780 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}))
48 coires1 6268 . . . . . 6 (𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) = (𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌}))
4948uneq1i 4158 . . . . 5 ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩})
5047, 49eqtrdi 2784 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}))
5116, 50syl3an1 1161 . . 3 ((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}))
5251f1oeq1d 6834 . 2 ((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴1-1-onto𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}):𝐴1-1-onto𝐵))
5315, 52mpbid 231 1 ((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}):𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3471  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4323  {csn 4629  {cpr 4631  cop 4635   I cid 5575   × cxp 5676  cres 5680  ccom 5682   Fn wfn 6543  1-1-ontowf1o 6547  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556
This theorem is referenced by:  dif1en  9184  dif1enOLD  9186
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