Theorem imass1 5005

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
imass1	⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Proof of Theorem imass1

Step	Hyp	Ref	Expression
1		ssres 4935	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))
2		rnss 4859	. . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
3	1, 2	syl 14	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶))
4		df-ima 4641	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
5		df-ima 4641	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
6	3, 4, 5	3sstr4g 3200	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶))

Syntax hints:   → wi 4   ⊆ wss 3131  ran crn 4629   ↾ cres 4630   “ cima 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by:  imasnopn  13838