Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fndmdifeq0 | Structured version Visualization version GIF version |
Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
fndmdifeq0 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndmdif 6913 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | |
2 | 1 | eqeq1d 2741 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅)) |
3 | rabeq0 4323 | . . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥)) | |
4 | nne 2948 | . . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)) | |
5 | 4 | ralbii 3092 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
6 | 3, 5 | bitri 274 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
7 | eqfnfv 6903 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
8 | 6, 7 | bitr4id 289 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ 𝐹 = 𝐺)) |
9 | 2, 8 | bitrd 278 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ≠ wne 2944 ∀wral 3065 {crab 3069 ∖ cdif 3888 ∅c0 4261 dom cdm 5588 Fn wfn 6425 ‘cfv 6430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-fv 6438 |
This theorem is referenced by: wemapso 9271 wemapso2lem 9272 |
Copyright terms: Public domain | W3C validator |