Theorem funimass2 3569

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 3567	. . . . 5 |- (Fun F -> (F"(`'F"B)) = (B i^i ran F))
2	1	sseq2d 2086	. . . 4 |- (Fun F -> ((F"A) (_ (F"(`'F"B)) <-> (F"A) (_ (B i^i ran F)))
3		inss1 2227	. . . . 5 |- (B i^i ran F) (_ B
4		sstr2 2068	. . . . 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 3429	. 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 2043   (_ wss 2044  `'ccnv 3165  ran crn 3167  "cima 3169  Fun wfun 3172

This theorem is referenced by:  fvimacnvi 3799

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188

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