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Mirrors > Home > MPE Home > Th. List > fndmdifcom | Structured version Visualization version GIF version |
Description: The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
fndmdifcom | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necom 2996 | . . 3 ⊢ ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥)) | |
2 | 1 | rabbii 3397 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)} |
3 | fndmdif 6901 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | |
4 | fndmdif 6901 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}) | |
5 | 4 | ancoms 458 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}) |
6 | 2, 3, 5 | 3eqtr4a 2805 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ≠ wne 2942 {crab 3067 ∖ cdif 3880 dom cdm 5580 Fn wfn 6413 ‘cfv 6418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-iota 6376 df-fun 6420 df-fn 6421 df-fv 6426 |
This theorem is referenced by: (None) |
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