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Mirrors > Home > MPE Home > Th. List > tpossym | Structured version Visualization version 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 8021 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐴) → tpos 𝐹 Fn (𝐴 × 𝐴)) | |
2 | eqfnov2 7362 | . . 3 ⊢ ((tpos 𝐹 Fn (𝐴 × 𝐴) ∧ 𝐹 Fn (𝐴 × 𝐴)) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦))) | |
3 | 1, 2 | mpancom 688 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦))) |
4 | eqcom 2746 | . . . 4 ⊢ ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦)) | |
5 | ovtpos 8007 | . . . . 5 ⊢ (𝑥tpos 𝐹𝑦) = (𝑦𝐹𝑥) | |
6 | 5 | eqeq2i 2752 | . . . 4 ⊢ ((𝑥𝐹𝑦) = (𝑥tpos 𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
7 | 4, 6 | bitri 278 | . . 3 ⊢ ((𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
8 | 7 | 2ralbii 3092 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥tpos 𝐹𝑦) = (𝑥𝐹𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
9 | 3, 8 | bitrdi 290 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐴) → (tpos 𝐹 = 𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝐹𝑦) = (𝑦𝐹𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∀wral 3064 × cxp 5567 Fn wfn 6396 (class class class)co 7235 tpos ctpos 7991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-fo 6407 df-fv 6409 df-ov 7238 df-tpos 7992 |
This theorem is referenced by: xmettpos 23279 |
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