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Theorem f1oprswap 6167
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 6164 . . . . 5 ((𝐴𝑉𝐴𝑉) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
21anidms 676 . . . 4 (𝐴𝑉 → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
32ad2antrr 761 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
4 dfsn2 4181 . . . . . 6 {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩}
5 opeq2 4394 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
6 opeq1 4393 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
75, 6preq12d 4267 . . . . . 6 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
84, 7syl5eq 2666 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
9 dfsn2 4181 . . . . . 6 {𝐴} = {𝐴, 𝐴}
10 preq2 4260 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
119, 10syl5eq 2666 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐴, 𝐵})
128, 11, 11f1oeq123d 6120 . . . 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 222 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
15 simpll 789 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝑉)
16 simplr 791 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐵𝑊)
17 simpr 477 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝐵)
18 fnprg 5935 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐵𝑊𝐴𝑉) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
1915, 16, 16, 15, 17, 18syl221anc 1335 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
20 cnvsng 5609 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21 cnvsng 5609 . . . . . . . . . 10 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2221ancoms 469 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2320, 22uneq12d 3760 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}))
24 uncom 3749 . . . . . . . 8 ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2523, 24syl6eq 2670 . . . . . . 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 4171 . . . . . . . 8 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2827cnveqi 5286 . . . . . . 7 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
29 cnvun 5526 . . . . . . 7 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3028, 29eqtri 2642 . . . . . 6 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3126, 30, 273eqtr4g 2679 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
3231fneq1d 5969 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
3319, 32mpbird 247 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
34 dff1o4 6132 . . 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 697 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
3614, 35pm2.61dane 2878 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  cun 3565  {csn 4168  {cpr 4170  cop 4174  ccnv 5103   Fn wfn 5871  1-1-ontowf1o 5875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883
This theorem is referenced by:  fveqf1o  6542  symg2bas  17799  subfacp1lem2a  31136
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