Theorem supp0cosupp0fn 6500

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

Assertion

Ref	Expression
supp0cosupp0fn	|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) )

Proof of Theorem supp0cosupp0fn

Step	Hyp	Ref	Expression
1		suppcofn 6499	. . 3 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
2		imaeq2 5120	. . . 4 |- ( ( F supp Z ) = (/) -> ( `' G " ( F supp Z ) ) = ( `' G " (/) ) )
3		ima0 5144	. . . 4 |- ( `' G " (/) ) = (/)
4	2, 3	eqtrdi 2287	. . 3 |- ( ( F supp Z ) = (/) -> ( `' G " ( F supp Z ) ) = (/) )
5	1, 4	sylan9eq 2291	. 2 |- ( ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) /\ ( F supp Z ) = (/) ) -> ( ( F o. G ) supp Z ) = (/) )
6	5	ex 115	1 |- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   (/)c0 3520   `'ccnv 4771   "cima 4775   o. ccom 4776   Fun wfun 5369  (class class class)co 6078   supp csupp 6468

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-nul 3521  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-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-supp 6469

This theorem is referenced by: (None)