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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 4231 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵) | |
2 | reseq2 5995 | . . . 4 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | sylbi 217 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵)) |
4 | 3 | rneqd 5952 | . 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵)) |
5 | df-ima 5702 | . . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵) | |
6 | resres 6013 | . . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
7 | 6 | rneqi 5951 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
8 | 5, 7 | eqtri 2763 | . 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵)) |
9 | df-ima 5702 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
10 | 4, 8, 9 | 3eqtr4g 2800 | 1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ⊆ wss 3963 ran crn 5690 ↾ cres 5691 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: ressuppss 8207 ressuppssdif 8209 naddcllem 8713 marypha1lem 9471 ackbij2lem3 10278 eqg0subgecsn 19228 dmdprdsplit2lem 20080 cnpresti 23312 cnprest 23313 limcflf 25931 limcresi 25935 limciun 25944 efopnlem2 26714 negsval 28072 pthhashvtx 35112 cvmopnlem 35263 cvmlift2lem9a 35288 poimirlem4 37611 limsupresre 45652 limsupresico 45656 liminfresico 45727 uhgrimisgrgric 47837 |
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