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Mirrors > Home > ILE Home > Th. List > fneqeql | GIF version |
Description: Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
fneqeql | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv 5486 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
2 | eqcom 2119 | . . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | |
3 | rabid2 2584 | . . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
5 | 1, 4 | syl6bbr 197 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴)) |
6 | fndmin 5495 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | |
7 | 6 | eqeq1d 2126 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∩ 𝐺) = 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴)) |
8 | 5, 7 | bitr4d 190 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∀wral 2393 {crab 2397 ∩ cin 3040 dom cdm 4509 Fn wfn 5088 ‘cfv 5093 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: fneqeql2 5497 fnreseql 5498 |
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