Theorem funimass1 3578

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 3577	. . . 4 |- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
2		dfss 2057	. . . . . 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 1483	. . . 4 |- (A (_ ran F -> (A i^i ran F) = A)
5	1, 4	sylan9eq 1530	. . 3 |- ((Fun F /\ A (_ ran F) -> (F"(`'F"A)) = A)
6	5	sseq1d 2091	. 2 |- ((Fun F /\ A (_ ran F) -> ((F"(`'F"A)) (_ (F"B) <-> A (_ (F"B)))
7		imass2 3439	. 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 958   i^i cin 2049   (_ wss 2050  `'ccnv 3175  ran crn 3177  "cima 3179  Fun wfun 3182

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

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