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Theorem tposfo 7548
Description: The domain and range of a transposition. (Contributed by NM, 10-Sep-2015.)
Assertion
Ref Expression
tposfo (𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)

Proof of Theorem tposfo
StepHypRef Expression
1 relxp 5283 . . 3 Rel (𝐴 × 𝐵)
2 tposfo2 7544 . . 3 (Rel (𝐴 × 𝐵) → (𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐴 × 𝐵)–onto𝐶))
31, 2ax-mp 5 . 2 (𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐴 × 𝐵)–onto𝐶)
4 cnvxp 5709 . . 3 (𝐴 × 𝐵) = (𝐵 × 𝐴)
5 foeq2 6273 . . 3 ((𝐴 × 𝐵) = (𝐵 × 𝐴) → (tpos 𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ tpos 𝐹:(𝐵 × 𝐴)–onto𝐶))
64, 5ax-mp 5 . 2 (tpos 𝐹:(𝐴 × 𝐵)–onto𝐶 ↔ tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)
73, 6sylib 208 1 (𝐹:(𝐴 × 𝐵)–onto𝐶 → tpos 𝐹:(𝐵 × 𝐴)–onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632   × cxp 5264  ccnv 5265  Rel wrel 5271  ontowfo 6047  tpos ctpos 7520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fo 6055  df-fv 6057  df-tpos 7521
This theorem is referenced by: (None)
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