Theorem funimass2 3573

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 3571	. . . . 5 |- (Fun F -> (F"(`'F"B)) = (B i^i ran F))
2	1	sseq2d 2089	. . . 4 |- (Fun F -> ((F"A) (_ (F"(`'F"B)) <-> (F"A) (_ (B i^i ran F)))
3		inss1 2230	. . . . 5 |- (B i^i ran F) (_ B
4		sstr2 2071	. . . . 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 3433	. 2 |- (A (_ (`'F"B) -> (F"A) (_ (F"(`'F"B)))
9	7, 8	sylan2 451	1 |- ((Fun F /\ A (_ (`'F"B)) -> (F"A) (_ B)

Syntax hints:   -> wi 3   /\ wa 223   i^i cin 2046   (_ wss 2047  `'ccnv 3169  ran crn 3171  "cima 3173  Fun wfun 3176

This theorem is referenced by:  fvimacnvi 3804

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192

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