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Mirrors > Home > MPE Home > Th. List > resima2 | Structured version Visualization version GIF version |
Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) |
Ref | Expression |
---|---|
resima2 | ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqin2 4142 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
2 | reseq2 5813 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | sylbi 220 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
4 | 3 | rneqd 5772 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵)) |
5 | df-ima 5532 | . . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵) | |
6 | resres 5831 | . . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
7 | 6 | rneqi 5771 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
8 | 5, 7 | eqtri 2821 | . 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
9 | df-ima 5532 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
10 | 4, 8, 9 | 3eqtr4g 2858 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 ran crn 5520 ↾ cres 5521 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: ressuppss 7832 ressuppssdif 7834 marypha1lem 8881 ackbij2lem3 9652 dmdprdsplit2lem 19160 cnpresti 21893 cnprest 21894 limcflf 24484 limcresi 24488 limciun 24497 efopnlem2 25248 pthhashvtx 32487 cvmopnlem 32638 cvmlift2lem9a 32663 poimirlem4 35061 limsupresre 42338 limsupresico 42342 liminfresico 42413 |
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