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Theorem xpcomf1o 8205
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 5266 . . . 4 Rel (𝐴 × 𝐵)
2 cnvf1o 7427 . . . 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 6268 . . . 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 221 . 2 𝐹:(𝐴 × 𝐵)–1-1-onto(𝐴 × 𝐵)
8 cnvxp 5692 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
9 f1oeq3 6270 . . 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 220 1 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  {csn 4316   cuni 4574  cmpt 4863   × cxp 5247  ccnv 5248  Rel wrel 5254  1-1-ontowf1o 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7315  df-2nd 7316
This theorem is referenced by:  xpcomco  8206  xpcomen  8207  omf1o  8219
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