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Theorem xpcomf1o 9012
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 5656 . . . 4 Rel (𝐴 × 𝐵)
2 cnvf1o 8048 . . . 4 (Rel (𝐴 × 𝐵) → (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵))
31, 2ax-mp 5 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
4 xpcomf1o.1 . . . 4 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})
5 f1oeq1 6777 . . . 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 5 . . 3 (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥}):(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵))
73, 6mpbir 230 . 2 𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
8 cnvxp 6114 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
9 f1oeq3 6779 . . 3 ((𝐴 × 𝐵) = (𝐵 × 𝐴) → (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)))
108, 9ax-mp 5 . 2 (𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴))
117, 10mpbi 229 1 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  {csn 4591   cuni 4870  cmpt 5193   × cxp 5636  ccnv 5637  Rel wrel 5643  1-1-ontowf1o 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1st 7926  df-2nd 7927
This theorem is referenced by:  xpcomco  9013  xpcomen  9014  omf1o  9026
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