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Mirrors > Home > MPE Home > Th. List > fun2ssres | Structured version Visualization version GIF version |
Description: Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
Ref | Expression |
---|---|
fun2ssres | ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resabs1 5764 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹 ↾ 𝐴)) | |
2 | 1 | eqcomd 2801 | . . 3 ⊢ (𝐴 ⊆ dom 𝐺 → (𝐹 ↾ 𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴)) |
3 | funssres 6268 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | |
4 | 3 | reseq1d 5733 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
5 | 2, 4 | sylan9eqr 2853 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
6 | 5 | 3impa 1103 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ⊆ wss 3859 dom cdm 5443 ↾ cres 5445 Fun wfun 6219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-res 5455 df-fun 6227 |
This theorem is referenced by: wfrlem12 7818 wfrlem14 7820 wfrlem17 7823 tfrlem9 7873 tfrlem9a 7874 tfrlem11 7876 bnj1503 31737 frrlem10 32741 frrlem12 32743 |
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