Theorem imacosuppfn 6502

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 6500	. . . 4 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
2	1	imaeq2d 5124	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( G " ( ( F o. G ) supp Z ) ) = ( G " ( `' G " ( F supp Z ) ) ) )
3		funforn 5620	. . . 4 |- ( Fun G <-> G : dom G -onto-> ran G )
4		foimacnv 5655	. . . 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 2291	. 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 1402   e. wcel 2209   C_ wss 3220   `'ccnv 4771   dom cdm 4772   ran crn 4773   "cima 4775   o. ccom 4776   Fun wfun 5369   -onto->wfo 5373  (class class class)co 6079   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by: (None)