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Theorem xpcomf1o 6299
 Description: The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypothesis
Ref Expression
xpcomf1o.1 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
Assertion
Ref Expression
xpcomf1o 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem xpcomf1o
StepHypRef Expression
1 relxp 4447 . . . 4 Rel (𝐴 × 𝐵)
2 cnvf1o 5846 . . . 4 (Rel (𝐴 × 𝐵) → (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵))
31, 2ax-mp 7 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
4 xpcomf1o.1 . . . 4 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
5 f1oeq1 5117 . . . 4 (𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}) → (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)))
64, 5ax-mp 7 . . 3 (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵))
73, 6mpbir 134 . 2 𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
8 cnvxp 4742 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
9 f1oeq3 5119 . . 3 ((𝐴 × 𝐵) = (𝐵 × 𝐴) → (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)))
108, 9ax-mp 7 . 2 (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴))
117, 10mpbi 133 1 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243  {csn 3375  ∪ cuni 3580   ↦ cmpt 3818   × cxp 4343  ◡ccnv 4344  Rel wrel 4350  –1-1-onto→wf1o 4901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by:  xpcomco  6300  xpcomen  6301
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