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Theorem xpcomf1o 6235
 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 4390 . . . 4 Rel (A × B)
2 cnvf1o 5788 . . . 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 5060 . . . 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 4685 . . 3 (A × B) = (B × A)
9 f1oeq3 5062 . . 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 1242  {csn 3367  ∪ cuni 3571   ↦ cmpt 3809   × cxp 4286  ◡ccnv 4287  Rel wrel 4293  –1-1-onto→wf1o 4844 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1st 5709  df-2nd 5710 This theorem is referenced by:  xpcomco  6236  xpcomen  6237
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