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Mirrors > Home > ILE Home > Th. List > fndmdifcom | 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 2392 | . . . 4 ⊢ ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥)) | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥))) |
3 | 2 | rabbiia 2671 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)} |
4 | fndmdif 5525 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | |
5 | fndmdif 5525 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}) | |
6 | 5 | ancoms 266 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}) |
7 | 3, 4, 6 | 3eqtr4a 2198 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 {crab 2420 ∖ cdif 3068 dom cdm 4539 Fn wfn 5118 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: (None) |
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