Theorem funimass1 3572

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

Assertion

Ref	Expression
funimass1	|- ((Fun F /\ A (_ ran F) -> ((`'F"A) (_ B -> A (_ (F"B)))

Proof of Theorem funimass1

Step	Hyp	Ref	Expression
1		funimacnv 3571	. . . 4 |- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
2		dfss 2054	. . . . . 6 |- (A (_ ran F <-> A = (A i^i ran F))
3	2	biimp 151	. . . . 5 |- (A (_ ran F -> A = (A i^i ran F))
4	3	eqcomd 1480	. . . 4 |- (A (_ ran F -> (A i^i ran F) = A)
5	1, 4	sylan9eq 1527	. . 3 |- ((Fun F /\ A (_ ran F) -> (F"(`'F"A)) = A)
6	5	sseq1d 2088	. 2 |- ((Fun F /\ A (_ ran F) -> ((F"(`'F"A)) (_ (F"B) <-> A (_ (F"B)))
7		imass2 3433	. 2 |- ((`'F"A) (_ B -> (F"(`'F"A)) (_ (F"B))
8	6, 7	syl5bi 208	1 |- ((Fun F /\ A (_ ran F) -> ((`'F"A) (_ B -> A (_ (F"B)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   i^i cin 2046   (_ wss 2047  `'ccnv 3169  ran crn 3171  "cima 3173  Fun wfun 3176

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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