Theorem imacosuppfn 6446

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 6444	. . . 4 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
2	1	imaeq2d 5082	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' G " ( F supp Z ) ) ) )
3		funforn 5575	. . . 4 |- ( Fun G <-> G : dom G -onto-> ran G )
4		foimacnv 5610	. . . 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 2284	. 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 2202   C_ wss 3201   `'ccnv 4730   dom cdm 4731   ran crn 4732   "cima 4734   o. ccom 4735   Fun wfun 5327   -onto->wfo 5331  (class class class)co 6028   supp csupp 6413

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by: (None)