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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 4189 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
2 | reseq2 5841 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | sylbi 218 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
4 | 3 | rneqd 5801 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵)) |
5 | df-ima 5561 | . . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵) | |
6 | resres 5859 | . . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
7 | 6 | rneqi 5800 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
8 | 5, 7 | eqtri 2841 | . 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
9 | df-ima 5561 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
10 | 4, 8, 9 | 3eqtr4g 2878 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∩ cin 3932 ⊆ wss 3933 ran crn 5549 ↾ cres 5550 “ cima 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: ressuppss 7838 ressuppssdif 7840 marypha1lem 8885 ackbij2lem3 9651 dmdprdsplit2lem 19096 cnpresti 21824 cnprest 21825 limcflf 24406 limcresi 24410 limciun 24419 efopnlem2 25167 pthhashvtx 32271 cvmopnlem 32422 cvmlift2lem9a 32447 poimirlem4 34777 limsupresre 41853 limsupresico 41857 liminfresico 41928 |
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