MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpossym Structured version   Visualization version   GIF version

Theorem tpossym 7586
Description: Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
tpossym (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem tpossym
StepHypRef Expression
1 tposfn 7583 . . 3 (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴))
2 eqfnov2 6964 . . 3 ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
31, 2mpancom 679 . 2 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
4 eqcom 2771 . . . 4 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦))
5 ovtpos 7569 . . . . 5 (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥)
65eqeq2i 2776 . . . 4 ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
74, 6bitri 266 . . 3 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
872ralbii 3127 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
93, 8syl6bb 278 1 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wral 3054   × cxp 5274   Fn wfn 6062  (class class class)co 6841  tpos ctpos 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-fo 6073  df-fv 6075  df-ov 6844  df-tpos 7554
This theorem is referenced by:  xmettpos  22432
  Copyright terms: Public domain W3C validator