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Theorem xpcomf1o 6327
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 4472 . . . 4 Rel (𝐴 × 𝐵)
2 cnvf1o 5871 . . . 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 5142 . . . 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 138 . 2 𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
8 cnvxp 4767 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
9 f1oeq3 5144 . . 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 137 1 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1257  {csn 3400   cuni 3605  cmpt 3843   × cxp 4368  ccnv 4369  Rel wrel 4375  1-1-ontowf1o 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-1st 5792  df-2nd 5793
This theorem is referenced by:  xpcomco  6328  xpcomen  6329
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