Theorem imass2 2622

Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.

Assertion

Ref	Expression
imass2	|- (A (_ B -> (C"A) (_ (C"B))

Proof of Theorem imass2

Step	Hyp	Ref	Expression
1		ssres2 2590	. . 3 |- (A (_ B -> (C |` A) (_ (C |` B))
2		rnss 2558	. . 3 |- ((C |` A) (_ (C |` B) -> ran (C |` A) (_ ran (C |` B))
3	1, 2	syl 12	. 2 |- (A (_ B -> ran (C |` A) (_ ran (C |` B))
4		df-ima 2431	. 2 |- (C"A) = ran (C |` A)
5		df-ima 2431	. 2 |- (C"B) = ran (C |` B)
6	3, 4, 5	3sstr4g 1541	1 |- (A (_ B -> (C"A) (_ (C"B))

Syntax hints:   -> wi 2   (_ wss 1487  ran crn 2411   |` cres 2412  "cima 2413

This theorem is referenced by:  funimass1 2712  funimass2 2713  sbthlem1 3349  sbthlem2 3350  php3 3411

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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