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Mirrors > Home > MPE Home > Th. List > fneqeql | Structured version Visualization version 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 6311 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
2 | eqcom 2629 | . . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | |
3 | rabid2 3118 | . . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) | |
4 | 2, 3 | bitri 264 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
5 | 1, 4 | syl6bbr 278 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴)) |
6 | fndmin 6324 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | |
7 | 6 | eqeq1d 2624 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∩ 𝐺) = 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)} = 𝐴)) |
8 | 5, 7 | bitr4d 271 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∀wral 2912 {crab 2916 ∩ cin 3573 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: fneqeql2 6326 fnreseql 6327 lspextmo 19056 |
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