Theorem imass2 5161

Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
imass2	⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Proof of Theorem imass2

Step	Hyp	Ref	Expression
1		ssres2 5088	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))
2		rnss 5010	. . 3 ⊢ ((𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵) → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵))
3	1, 2	syl 14	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐶 ↾ 𝐴) ⊆ ran (𝐶 ↾ 𝐵))
4		df-ima 4785	. 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
5		df-ima 4785	. 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵)
6	3, 4, 5	3sstr4g 3291	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Syntax hints:   → wi 4   ⊆ wss 3220  ran crn 4773   ↾ cres 4774   “ cima 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by:  funimass1  5456  funimass2  5457  fvimacnv  5818  fnfvimad  5948  f1imass  5974  ecinxp  6878  sbthlem1  7268  sbthlem2  7269  iscnp4  15302  cnptopco  15306  cnntri  15308  cnrest2  15320  cnptopresti  15322  cnptoprest  15323  metcnp3  15595