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Theorem tpossym 5813
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 = 𝐹x A y A (x𝐹y) = (y𝐹x)))
Distinct variable groups:   x,y,A   x,𝐹,y

Proof of Theorem tpossym
StepHypRef Expression
1 tposfn 5810 . . 3 (𝐹 Fn (A × A) → tpos 𝐹 Fn (A × A))
2 eqfnov2 5531 . . 3 ((tpos 𝐹 Fn (A × A) 𝐹 Fn (A × A)) → (tpos 𝐹 = 𝐹x A y A (xtpos 𝐹y) = (x𝐹y)))
31, 2mpancom 401 . 2 (𝐹 Fn (A × A) → (tpos 𝐹 = 𝐹x A y A (xtpos 𝐹y) = (x𝐹y)))
4 eqcom 2024 . . . 4 ((xtpos 𝐹y) = (x𝐹y) ↔ (x𝐹y) = (xtpos 𝐹y))
5 vex 2538 . . . . . 6 x V
6 vex 2538 . . . . . 6 y V
7 ovtposg 5796 . . . . . 6 ((x V y V) → (xtpos 𝐹y) = (y𝐹x))
85, 6, 7mp2an 404 . . . . 5 (xtpos 𝐹y) = (y𝐹x)
98eqeq2i 2032 . . . 4 ((x𝐹y) = (xtpos 𝐹y) ↔ (x𝐹y) = (y𝐹x))
104, 9bitri 173 . . 3 ((xtpos 𝐹y) = (x𝐹y) ↔ (x𝐹y) = (y𝐹x))
11102ralbii 2310 . 2 (x A y A (xtpos 𝐹y) = (x𝐹y) ↔ x A y A (x𝐹y) = (y𝐹x))
123, 11syl6bb 185 1 (𝐹 Fn (A × A) → (tpos 𝐹 = 𝐹x A y A (x𝐹y) = (y𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  wral 2284  Vcvv 2535   × cxp 4270   Fn wfn 4824  (class class class)co 5436  tpos ctpos 5781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-ov 5439  df-tpos 5782
This theorem is referenced by: (None)
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