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Theorem f1oprswap 6893
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 6890 . . . . 5 ((𝐴𝑉𝐴𝑉) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
21anidms 566 . . . 4 (𝐴𝑉 → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
32ad2antrr 726 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
4 dfsn2 4644 . . . . . 6 {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩}
5 opeq2 4879 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
6 opeq1 4878 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
75, 6preq12d 4746 . . . . . 6 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
84, 7eqtrid 2787 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
9 dfsn2 4644 . . . . . 6 {𝐴} = {𝐴, 𝐴}
10 preq2 4739 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
119, 10eqtrid 2787 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐴, 𝐵})
128, 11, 11f1oeq123d 6843 . . . 4 (𝐴 = 𝐵 → ({⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵}))
1312adantl 481 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → ({⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵}))
143, 13mpbid 232 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
15 simpll 767 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝑉)
16 simplr 769 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐵𝑊)
17 simpr 484 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝐵)
18 fnprg 6627 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐵𝑊𝐴𝑉) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
1915, 16, 16, 15, 17, 18syl221anc 1380 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
20 cnvsng 6245 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21 cnvsng 6245 . . . . . . . . . 10 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2221ancoms 458 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2320, 22uneq12d 4179 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}))
24 uncom 4168 . . . . . . . 8 ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2523, 24eqtrdi 2791 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
2625adantr 480 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
27 df-pr 4634 . . . . . . . 8 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2827cnveqi 5888 . . . . . . 7 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
29 cnvun 6165 . . . . . . 7 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3028, 29eqtri 2763 . . . . . 6 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3126, 30, 273eqtr4g 2800 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
3231fneq1d 6662 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
3319, 32mpbird 257 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
34 dff1o4 6857 . . 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 3027 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  cun 3961  {csn 4631  {cpr 4633  cop 4637  ccnv 5688   Fn wfn 6558  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  fveqf1o  7322  f1ofvswap  7326  symg2bas  19425  subfacp1lem2a  35165
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