Theorem imasaddflemg 12737

Description: The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
imasaddf.f	|- ( ph -> F : V -onto-> B )
imasaddf.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
imasaddflem.a	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
imasaddfnlemg.v	|- ( ph -> V e. W )
imasaddfnlemg.x	|- ( ph -> .x. e. C )
imasaddflem.c	|- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )

Assertion

Ref	Expression
imasaddflemg	|- ( ph -> .xb : ( B X. B ) --> B )

Distinct variable groups:   q, p, B   a, b, p, q, V   .x. , p, q   F, a, b, p, q   ph, a, b, p, q   .xb , a, b, p, q

Allowed substitution hints:   B( a, b)   C( q, p, a, b)   .x. ( a, b)   W( q, p, a, b)

Proof of Theorem imasaddflemg

Step	Hyp	Ref	Expression
1		imasaddf.f	. . 3 |- ( ph -> F : V -onto-> B )
2		imasaddf.e	. . 3 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )
3		imasaddflem.a	. . 3 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
4		imasaddfnlemg.v	. . 3 |- ( ph -> V e. W )
5		imasaddfnlemg.x	. . 3 |- ( ph -> .x. e. C )
6	1, 2, 3, 4, 5	imasaddfnlemg 12735	. 2 |- ( ph -> .xb Fn ( B X. B ) )
7		fof 5439	. . . . . . . . . 10 |- ( F : V -onto-> B -> F : V --> B )
8	1, 7	syl 14	. . . . . . . . 9 |- ( ph -> F : V --> B )
9		ffvelcdm 5650	. . . . . . . . . . 11 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
10		ffvelcdm 5650	. . . . . . . . . . 11 |- ( ( F : V --> B /\ q e. V ) -> ( F ` q ) e. B )
11	9, 10	anim12dan 600	. . . . . . . . . 10 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> ( ( F ` p ) e. B /\ ( F ` q ) e. B ) )
12		opelxpi 4659	. . . . . . . . . 10 |- ( ( ( F ` p ) e. B /\ ( F ` q ) e. B ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
13	11, 12	syl 14	. . . . . . . . 9 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
14	8, 13	sylan 283	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
15		imasaddflem.c	. . . . . . . . 9 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
16		ffvelcdm 5650	. . . . . . . . 9 |- ( ( F : V --> B /\ ( p .x. q ) e. V ) -> ( F ` ( p .x. q ) ) e. B )
17	8, 15, 16	syl2an2r 595	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( F ` ( p .x. q ) ) e. B )
18	14, 17	opelxpd 4660	. . . . . . 7 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. ( ( B X. B ) X. B ) )
19	18	snssd 3738	. . . . . 6 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
20	19	anassrs 400	. . . . 5 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
21	20	iunssd 3933	. . . 4 |- ( ( ph /\ p e. V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
22	21	iunssd 3933	. . 3 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
23	3, 22	eqsstrd 3192	. 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
24		dff2 5661	. 2 |- ( .xb : ( B X. B ) --> B <-> ( .xb Fn ( B X. B ) /\ .xb C_ ( ( B X. B ) X. B ) ) )
25	6, 23, 24	sylanbrc 417	1 |- ( ph -> .xb : ( B X. B ) --> B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3130   {csn 3593   <.cop 3596   U_ciun 3887   X. cxp 4625   Fn wfn 5212   -->wf 5213   -onto->wfo 5215   ` cfv 5217  (class class class)co 5875

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878

This theorem is referenced by:  imasaddf  12740  imasmulf  12743  qusaddflemg  12753