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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 2988 | . . 3 ⊢ ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥)) | |
2 | 1 | rabbii 3432 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)} |
3 | fndmdif 7037 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | |
4 | fndmdif 7037 | . . 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 2792 | 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 1533 ≠ wne 2934 {crab 3426 ∖ cdif 3940 dom cdm 5669 Fn wfn 6532 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fn 6540 df-fv 6545 |
This theorem is referenced by: (None) |
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