Theorem resima2 5971

Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.)

Assertion

Ref	Expression
resima2	⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Proof of Theorem resima2

Step	Hyp	Ref	Expression
1		sseqin2 4172	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
2		reseq2 5929	. . . 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 5884	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵))
5		df-ima 5634	. . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
6		resres 5947	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
7	6	rneqi 5883	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ↾ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
8	5, 7	eqtri 2756	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
9		df-ima 5634	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
10	4, 8, 9	3eqtr4g 2793	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3897   ⊆ wss 3898  ran crn 5622   ↾ cres 5623   “ cima 5624

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by:  ressuppss  8121  ressuppssdif  8123  naddcllem  8599  marypha1lem  9326  ackbij2lem3  10140  eqg0subgecsn  19113  dmdprdsplit2lem  19963  cnpresti  23206  cnprest  23207  limcflf  25812  limcresi  25816  limciun  25825  efopnlem2  26596  negsval  27970  pthhashvtx  35195  cvmopnlem  35345  cvmlift2lem9a  35370  poimirlem4  37687  limsupresre  45821  limsupresico  45825  liminfresico  45896  uhgrimisgrgric  48058