Theorem funimass1 2712

Description: A kind of contraposition law that infers a subclass of an image from a converse 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 2711	. . . 4 |- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
2		dfss 1493	. . . . . 6 |- (A (_ ran F <-> A = (A i^i ran F))
3	2	biimp 133	. . . . 5 |- (A (_ ran F -> A = (A i^i ran F))
4	3	cleqcomd 1106	. . . 4 |- (A (_ ran F -> (A i^i ran F) = A)
5	1, 4	sylan9eq 1144	. . 3 |- ((Fun F /\ A (_ ran F) -> (F"(`'F"A)) = A)
6	5	sseq1d 1527	. 2 |- ((Fun F /\ A (_ ran F) -> ((F"(`'F"A)) (_ (F"B) <-> A (_ (F"B)))
7		imass2 2622	. 2 |- ((`'F"A) (_ B -> (F"(`'F"A)) (_ (F"B))
8	6, 7	syl5bi 183	1 |- ((Fun F /\ A (_ ran F) -> ((`'F"A) (_ B -> A (_ (F"B)))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   i^i cin 1486   (_ wss 1487  `'ccnv 2409  ran crn 2411  "cima 2413  Fun wfun 2416

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

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