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Mirrors > Home > ILE Home > Th. List > tpossym | GIF version |
Description: Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tpossym | ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfn 5888 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴)) | |
2 | eqfnov2 5608 | . . 3 ⊢ ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦))) | |
3 | 1, 2 | mpancom 399 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦))) |
4 | eqcom 2042 | . . . 4 ⊢ ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦)) | |
5 | vex 2560 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | vex 2560 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | ovtposg 5874 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥)) | |
8 | 5, 6, 7 | mp2an 402 | . . . . 5 ⊢ (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥) |
9 | 8 | eqeq2i 2050 | . . . 4 ⊢ ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
10 | 4, 9 | bitri 173 | . . 3 ⊢ ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
11 | 10 | 2ralbii 2332 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
12 | 3, 11 | syl6bb 185 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 Vcvv 2557 × cxp 4343 Fn wfn 4897 (class class class)co 5512 tpos ctpos 5859 |
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 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-ov 5515 df-tpos 5860 |
This theorem is referenced by: (None) |
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