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Theorem tpossym 6301
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 6298 . . 3 (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴))
2 eqfnov2 6004 . . 3 ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
31, 2mpancom 422 . 2 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦)))
4 eqcom 2191 . . . 4 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦))
5 vex 2755 . . . . . 6 𝑥 ∈ V
6 vex 2755 . . . . . 6 𝑦 ∈ V
7 ovtposg 6284 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥))
85, 6, 7mp2an 426 . . . . 5 (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥)
98eqeq2i 2200 . . . 4 ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
104, 9bitri 184 . . 3 ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
11102ralbii 2498 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
123, 11bitrdi 196 1 (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752   × cxp 4642   Fn wfn 5230  (class class class)co 5896  tpos ctpos 6269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243  df-ov 5899  df-tpos 6270
This theorem is referenced by:  xmettpos  14327
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