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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 4192 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
2 | reseq2 5848 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | sylbi 219 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
4 | 3 | rneqd 5808 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵)) |
5 | df-ima 5568 | . . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵) | |
6 | resres 5866 | . . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
7 | 6 | rneqi 5807 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
8 | 5, 7 | eqtri 2844 | . 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
9 | df-ima 5568 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
10 | 4, 8, 9 | 3eqtr4g 2881 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3935 ⊆ wss 3936 ran crn 5556 ↾ cres 5557 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: ressuppss 7849 ressuppssdif 7851 marypha1lem 8897 ackbij2lem3 9663 dmdprdsplit2lem 19167 cnpresti 21896 cnprest 21897 limcflf 24479 limcresi 24483 limciun 24492 efopnlem2 25240 pthhashvtx 32374 cvmopnlem 32525 cvmlift2lem9a 32550 poimirlem4 34911 limsupresre 42026 limsupresico 42030 liminfresico 42101 |
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