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Theorem f1oprswap 6654
Description: A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
Assertion
Ref Expression
f1oprswap ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})

Proof of Theorem f1oprswap
StepHypRef Expression
1 f1osng 6651 . . . . 5 ((𝐴𝑉𝐴𝑉) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
21anidms 567 . . . 4 (𝐴𝑉 → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
32ad2antrr 722 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
4 dfsn2 4576 . . . . . 6 {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩}
5 opeq2 4802 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
6 opeq1 4801 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
75, 6preq12d 4675 . . . . . 6 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
84, 7syl5eq 2872 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
9 dfsn2 4576 . . . . . 6 {𝐴} = {𝐴, 𝐴}
10 preq2 4668 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
119, 10syl5eq 2872 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐴, 𝐵})
128, 11, 11f1oeq123d 6606 . . . 4 (𝐴 = 𝐵 → ({⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵}))
1312adantl 482 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → ({⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵}))
143, 13mpbid 233 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
15 simpll 763 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝑉)
16 simplr 765 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐵𝑊)
17 simpr 485 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝐵)
18 fnprg 6409 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐵𝑊𝐴𝑉) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
1915, 16, 16, 15, 17, 18syl221anc 1375 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
20 cnvsng 6077 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21 cnvsng 6077 . . . . . . . . . 10 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2221ancoms 459 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2320, 22uneq12d 4143 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}))
24 uncom 4132 . . . . . . . 8 ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2523, 24syl6eq 2876 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
2625adantr 481 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
27 df-pr 4566 . . . . . . . 8 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2827cnveqi 5743 . . . . . . 7 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
29 cnvun 5998 . . . . . . 7 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3028, 29eqtri 2848 . . . . . 6 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3126, 30, 273eqtr4g 2885 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
3231fneq1d 6442 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
3319, 32mpbird 258 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
34 dff1o4 6619 . . 3 ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵} ↔ ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ∧ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
3519, 33, 34sylanbrc 583 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
3614, 35pm2.61dane 3108 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wne 3020  cun 3937  {csn 4563  {cpr 4565  cop 4569  ccnv 5552   Fn wfn 6346  1-1-ontowf1o 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358
This theorem is referenced by:  fveqf1o  7055  symg2bas  18449  subfacp1lem2a  32314
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