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Mirrors > Home > MPE Home > Th. List > resresdm | Structured version Visualization version GIF version |
Description: A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.) |
Ref | Expression |
---|---|
resresdm | ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ 𝐴)) | |
2 | dmeq 5863 | . . . 4 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → dom 𝐹 = dom (𝐸 ↾ 𝐴)) | |
3 | 2 | reseq2d 5941 | . . 3 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸 ↾ 𝐴))) |
4 | resdmres 6188 | . . 3 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴) | |
5 | 3, 4 | eqtr2di 2790 | . 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ 𝐴) = (𝐸 ↾ dom 𝐹)) |
6 | 1, 5 | eqtrd 2773 | 1 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 dom cdm 5637 ↾ cres 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 |
This theorem is referenced by: uhgrspan1 28300 |
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