Theorem imacosuppfn 6467

Description: The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)

Assertion

Ref	Expression
imacosuppfn	|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )

Proof of Theorem imacosuppfn

Step	Hyp	Ref	Expression
1		suppcofn 6465	. . . 4 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
2	1	imaeq2d 5100	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' G " ( F supp Z ) ) ) )
3		funforn 5596	. . . 4 |- ( Fun G <-> G : dom G -onto-> ran G )
4		foimacnv 5631	. . . 4 |- ( ( G : dom G -onto-> ran G /\ ( F supp Z ) C_ ran G ) -> ( G " ( `' G " ( F supp Z ) ) ) = ( F supp Z ) )
5	3, 4	sylanb 284	. . 3 |- ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( `' G " ( F supp Z ) ) ) = ( F supp Z ) )
6	2, 5	sylan9eq 2285	. 2 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ ( Fun G /\ ( F supp Z ) C_ ran G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) )
7	6	ex 115	1 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   C_ wss 3210   `'ccnv 4747   dom cdm 4748   ran crn 4749   "cima 4751   o. ccom 4752   Fun wfun 5345   -onto->wfo 5349  (class class class)co 6049   supp csupp 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by: (None)