Theorem resima2 5642

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 4015	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐶 ∩ 𝐵) = 𝐵)
2		reseq2 5595	. . . 4 ⊢ ((𝐶 ∩ 𝐵) = 𝐵 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
3	1, 2	sylbi 209	. . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ↾ (𝐶 ∩ 𝐵)) = (𝐴 ↾ 𝐵))
4	3	rneqd 5556	. 2 ⊢ (𝐵 ⊆ 𝐶 → ran (𝐴 ↾ (𝐶 ∩ 𝐵)) = ran (𝐴 ↾ 𝐵))
5		df-ima 5325	. . 3 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran ((𝐴 ↾ 𝐶) ↾ 𝐵)
6		resres 5620	. . . 4 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵))
7	6	rneqi 5555	. . 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 5325	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
10	4, 8, 9	3eqtr4g 2858	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1653   ∩ cin 3768   ⊆ wss 3769  ran crn 5313   ↾ cres 5314   “ cima 5315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325

This theorem is referenced by:  ressuppss  7551  ressuppssdif  7553  marypha1lem  8581  ackbij2lem3  9351  dmdprdsplit2lem  18760  cnpresti  21421  cnprest  21422  limcflf  23986  limcresi  23990  limciun  23999  efopnlem2  24744  cvmopnlem  31777  cvmlift2lem9a  31802  poimirlem4  33902  limsupresre  40672  limsupresico  40676  liminfresico  40747