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Theorem f1oprswap 6092
 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 6089 . . . . 5 ((𝐴𝑉𝐴𝑉) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
21anidms 675 . . . 4 (𝐴𝑉 → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
32ad2antrr 758 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐴⟩}:{𝐴}–1-1-onto→{𝐴})
4 dfsn2 4138 . . . . . 6 {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩}
5 opeq2 4341 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
6 opeq1 4340 . . . . . . 7 (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
75, 6preq12d 4220 . . . . . 6 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
84, 7syl5eq 2656 . . . . 5 (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
9 dfsn2 4138 . . . . . 6 {𝐴} = {𝐴, 𝐴}
10 preq2 4213 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
119, 10syl5eq 2656 . . . . 5 (𝐴 = 𝐵 → {𝐴} = {𝐴, 𝐵})
128, 11, 11f1oeq123d 6046 . . . 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 221 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
15 simpll 786 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝑉)
16 simplr 788 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐵𝑊)
17 simpr 476 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → 𝐴𝐵)
18 fnprg 5861 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐵𝑊𝐴𝑉) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
1915, 16, 16, 15, 17, 18syl221anc 1329 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
20 cnvsng 5539 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21 cnvsng 5539 . . . . . . . . . 10 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2221ancoms 468 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
2320, 22uneq12d 3730 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}))
24 uncom 3719 . . . . . . . 8 ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2523, 24syl6eq 2660 . . . . . . 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 4128 . . . . . . . 8 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
2827cnveqi 5219 . . . . . . 7 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
29 cnvun 5457 . . . . . . 7 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3028, 29eqtri 2632 . . . . . 6 {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
3126, 30, 273eqtr4g 2669 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
3231fneq1d 5895 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
3319, 32mpbird 246 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
34 dff1o4 6058 . . 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 695 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
3614, 35pm2.61dane 2869 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131  ◡ccnv 5037   Fn wfn 5799  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  fveqf1o  6457  symg2bas  17641  subfacp1lem2a  30416
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