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 4176	. . . 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 5636	. . 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 2752	. 2 ⊢ ((𝐴 ↾ 𝐶) “ 𝐵) = ran (𝐴 ↾ (𝐶 ∩ 𝐵))
9		df-ima 5636	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
10	4, 8, 9	3eqtr4g 2789	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) “ 𝐵) = (𝐴 “ 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3904   ⊆ wss 3905  ran crn 5624   ↾ cres 5625   “ cima 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  ressuppss  8123  ressuppssdif  8125  naddcllem  8601  marypha1lem  9342  ackbij2lem3  10153  eqg0subgecsn  19094  dmdprdsplit2lem  19944  cnpresti  23191  cnprest  23192  limcflf  25798  limcresi  25802  limciun  25811  efopnlem2  26582  negsval  27954  pthhashvtx  35103  cvmopnlem  35253  cvmlift2lem9a  35278  poimirlem4  37606  limsupresre  45681  limsupresico  45685  liminfresico  45756  uhgrimisgrgric  47919