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Theorem tpossym 5995
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 5992 . . 3 (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴))
2 eqfnov2 5709 . . 3 ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
31, 2mpancom 413 . 2 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
4 eqcom 2087 . . . 4 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦))
5 vex 2618 . . . . . 6 𝑥 ∈ V
6 vex 2618 . . . . . 6 𝑦 ∈ V
7 ovtposg 5978 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥))
85, 6, 7mp2an 417 . . . . 5 (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥)
98eqeq2i 2095 . . . 4 ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
104, 9bitri 182 . . 3 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
11102ralbii 2382 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
123, 11syl6bb 194 1 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wcel 1436  wral 2355  Vcvv 2615   × cxp 4409   Fn wfn 4976  (class class class)co 5613  tpos ctpos 5963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fo 4987  df-fv 4989  df-ov 5616  df-tpos 5964
This theorem is referenced by: (None)
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