Theorem imasaddflemg 13349

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 13347	. 2 |- ( ph -> .xb Fn ( B X. B ) )
7		fof 5548	. . . . . . . . . 10 |- ( F : V -onto-> B -> F : V --> B )
8	1, 7	syl 14	. . . . . . . . 9 |- ( ph -> F : V --> B )
9		ffvelcdm 5768	. . . . . . . . . . 11 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
10		ffvelcdm 5768	. . . . . . . . . . 11 |- ( ( F : V --> B /\ q e. V ) -> ( F ` q ) e. B )
11	9, 10	anim12dan 602	. . . . . . . . . 10 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> ( ( F ` p ) e. B /\ ( F ` q ) e. B ) )
12		opelxpi 4751	. . . . . . . . . 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 5768	. . . . . . . . 9 |- ( ( F : V --> B /\ ( p .x. q ) e. V ) -> ( F ` ( p .x. q ) ) e. B )
17	8, 15, 16	syl2an2r 597	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( F ` ( p .x. q ) ) e. B )
18	14, 17	opelxpd 4752	. . . . . . 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 3813	. . . . . 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 4011	. . . 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 4011	. . 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 3260	. 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
24		dff2 5779	. 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 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   {csn 3666   <.cop 3669   U_ciun 3965   X. cxp 4717   Fn wfn 5313   -->wf 5314   -onto->wfo 5316   ` cfv 5318  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004

This theorem is referenced by:  imasaddf  13352  imasmulf  13355  qusaddflemg  13367