Theorem funimass2 3579

Description: A kind of contraposition law that infers an image subclass from a subclass of a pre-image.

Assertion

Ref	Expression
funimass2	|- ((Fun F /\ A (_ (`'F"B)) -> (F"A) (_ B)

Proof of Theorem funimass2

Step	Hyp	Ref	Expression
1		funimacnv 3577	. . . . 5 |- (Fun F -> (F"(`'F"B)) = (B i^i ran F))
2	1	sseq2d 2092	. . . 4 |- (Fun F -> ((F"A) (_ (F"(`'F"B)) <-> (F"A) (_ (B i^i ran F)))
3		inss1 2233	. . . . 5 |- (B i^i ran F) (_ B
4		sstr2 2074	. . . . 5 |- ((F"A) (_ (B i^i ran F) -> ((B i^i ran F) (_ B -> (F"A) (_ B))
5	3, 4	mpi 44	. . . 4 |- ((F"A) (_ (B i^i ran F) -> (F"A) (_ B)
6	2, 5	syl6bi 214	. . 3 |- (Fun F -> ((F"A) (_ (F"(`'F"B)) -> (F"A) (_ B))
7	6	imp 350	. 2 |- ((Fun F /\ (F"A) (_ (F"(`'F"B))) -> (F"A) (_ B)
8		imass2 3439	. 2 |- (A (_ (`'F"B) -> (F"A) (_ (F"(`'F"B)))
9	7, 8	sylan2 453	1 |- ((Fun F /\ A (_ (`'F"B)) -> (F"A) (_ B)

Syntax hints:   -> wi 3   /\ wa 223   i^i cin 2049   (_ wss 2050  `'ccnv 3175  ran crn 3177  "cima 3179  Fun wfun 3182

This theorem is referenced by:  fvimacnvi 3810

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

Copyright terms: Public domain