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Mirrors > Home > MPE Home > Th. List > fvreseq0 | Structured version Visualization version GIF version |
Description: Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.) |
Ref | Expression |
---|---|
fvreseq0 | ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnssres 6464 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) | |
2 | fnssres 6464 | . . 3 ⊢ ((𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ↾ 𝐵) Fn 𝐵) | |
3 | eqfnfv 6796 | . . . 4 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥))) | |
4 | fvres 6683 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) | |
5 | fvres 6683 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥)) | |
6 | 4, 5 | eqeq12d 2837 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 6 | ralbiia 3164 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) |
8 | 3, 7 | syl6bb 289 | . . 3 ⊢ (((𝐹 ↾ 𝐵) Fn 𝐵 ∧ (𝐺 ↾ 𝐵) Fn 𝐵) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
9 | 1, 2, 8 | syl2an 597 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝐺 Fn 𝐶 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
10 | 9 | an4s 658 | 1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ↾ cres 5551 Fn wfn 6344 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: fvreseq1 6803 fvreseq 6804 limsupequzlem 41996 |
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