Theorem suppun 7315

Description: The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)

Hypothesis

Ref	Expression
suppun.g	|- ( ph -> G e. V )

Assertion

Ref	Expression
suppun	|- ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )

Proof of Theorem suppun

Step	Hyp	Ref	Expression
1		ssun1 3776	. . . . . 6 |- ( `' F " ( _V \ { Z } ) ) C_ ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
2		cnvun 5538	. . . . . . . 8 |- `' ( F u. G ) = ( `' F u. `' G )
3	2	imaeq1i 5463	. . . . . . 7 |- ( `' ( F u. G ) " ( _V \ { Z } ) ) = ( ( `' F u. `' G ) " ( _V \ { Z } ) )
4		imaundir 5546	. . . . . . 7 |- ( ( `' F u. `' G ) " ( _V \ { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
5	3, 4	eqtri 2644	. . . . . 6 |- ( `' ( F u. G ) " ( _V \ { Z } ) ) = ( ( `' F " ( _V \ { Z } ) ) u. ( `' G " ( _V \ { Z } ) ) )
6	1, 5	sseqtr4i 3638	. . . . 5 |- ( `' F " ( _V \ { Z } ) ) C_ ( `' ( F u. G ) " ( _V \ { Z } ) )
7	6	a1i 11	. . . 4 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( `' F " ( _V \ { Z } ) ) C_ ( `' ( F u. G ) " ( _V \ { Z } ) ) )
8		suppimacnv 7306	. . . . 5 |- ( ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
9	8	adantr 481	. . . 4 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
10		suppun.g	. . . . . 6 |- ( ph -> G e. V )
11		unexg 6959	. . . . . . 7 |- ( ( F e. _V /\ G e. V ) -> ( F u. G ) e. _V )
12	11	adantlr 751	. . . . . 6 |- ( ( ( F e. _V /\ Z e. _V ) /\ G e. V ) -> ( F u. G ) e. _V )
13	10, 12	sylan2 491	. . . . 5 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F u. G ) e. _V )
14		simplr 792	. . . . 5 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> Z e. _V )
15		suppimacnv 7306	. . . . 5 |- ( ( ( F u. G ) e. _V /\ Z e. _V ) -> ( ( F u. G ) supp Z ) = ( `' ( F u. G ) " ( _V \ { Z } ) ) )
16	13, 14, 15	syl2anc 693	. . . 4 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( ( F u. G ) supp Z ) = ( `' ( F u. G ) " ( _V \ { Z } ) ) )
17	7, 9, 16	3sstr4d 3648	. . 3 |- ( ( ( F e. _V /\ Z e. _V ) /\ ph ) -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )
18	17	ex 450	. 2 |- ( ( F e. _V /\ Z e. _V ) -> ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) )
19		supp0prc 7298	. . . 4 |- ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) = (/) )
20		0ss 3972	. . . 4 |- (/) C_ ( ( F u. G ) supp Z )
21	19, 20	syl6eqss 3655	. . 3 |- ( -. ( F e. _V /\ Z e. _V ) -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )
22	21	a1d 25	. 2 |- ( -. ( F e. _V /\ Z e. _V ) -> ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) )
23	18, 22	pm2.61i 176	1 |- ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   `'ccnv 5113   "cima 5117  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  fsuppunbi  8296  gsumzaddlem  18321