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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 5571 | . . . 4 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → dom 𝐹 = dom (𝐸 ↾ 𝐴)) | |
3 | 2 | reseq2d 5644 | . . 3 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ dom 𝐹) = (𝐸 ↾ dom (𝐸 ↾ 𝐴))) |
4 | resdmres 5881 | . . 3 ⊢ (𝐸 ↾ dom (𝐸 ↾ 𝐴)) = (𝐸 ↾ 𝐴) | |
5 | 3, 4 | syl6req 2831 | . 2 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → (𝐸 ↾ 𝐴) = (𝐸 ↾ dom 𝐹)) |
6 | 1, 5 | eqtrd 2814 | 1 ⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 dom cdm 5357 ↾ cres 5359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-xp 5363 df-rel 5364 df-cnv 5365 df-dm 5367 df-rn 5368 df-res 5369 |
This theorem is referenced by: uhgrspan1 26667 |
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