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Theorem xpcomf1o 6541
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 4547 . . . 4 Rel (𝐴 × 𝐵)
2 cnvf1o 5990 . . . 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 5244 . . . 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 144 . 2 𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
8 cnvxp 4850 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
9 f1oeq3 5246 . . 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 143 1 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1289  {csn 3446   cuni 3653  cmpt 3899   × cxp 4436  ccnv 4437  Rel wrel 4443  1-1-ontowf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  xpcomco  6542  xpcomen  6543
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