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Theorem tpossym 7271
 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 7268 . . 3 (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴))
2 eqfnov2 6665 . . 3 ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
31, 2mpancom 700 . 2 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
4 eqcom 2617 . . . 4 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦))
5 ovtpos 7254 . . . . 5 (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥)
65eqeq2i 2622 . . . 4 ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
74, 6bitri 263 . . 3 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
872ralbii 2964 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
93, 8syl6bb 275 1 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∀wral 2896   × cxp 5036   Fn wfn 5799  (class class class)co 6549  tpos ctpos 7238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-tpos 7239 This theorem is referenced by:  xmettpos  21964
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