Theorem imass2 5114

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 5042	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))
2		rnss 4964	. . 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 4740	. 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
5		df-ima 4740	. 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵)
6	3, 4, 5	3sstr4g 3269	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵))

Syntax hints:   → wi 4   ⊆ wss 3199  ran crn 4728   ↾ cres 4729   “ cima 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-xp 4733  df-cnv 4735  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740

This theorem is referenced by:  funimass1  5409  funimass2  5410  fvimacnv  5765  fnfvimad  5895  f1imass  5920  ecinxp  6784  sbthlem1  7161  sbthlem2  7162  iscnp4  14971  cnptopco  14975  cnntri  14977  cnrest2  14989  cnptopresti  14991  cnptoprest  14992  metcnp3  15264