Theorem imasaddflemg 13637

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 13635	. 2 |- ( ph -> .xb Fn ( B X. B ) )
7		fof 5615	. . . . . . . . . 10 |- ( F : V -onto-> B -> F : V --> B )
8	1, 7	syl 14	. . . . . . . . 9 |- ( ph -> F : V --> B )
9		ffvelcdm 5841	. . . . . . . . . . 11 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
10		ffvelcdm 5841	. . . . . . . . . . 11 |- ( ( F : V --> B /\ q e. V ) -> ( F ` q ) e. B )
11	9, 10	anim12dan 608	. . . . . . . . . 10 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> ( ( F ` p ) e. B /\ ( F ` q ) e. B ) )
12		opelxpi 4806	. . . . . . . . . 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 5841	. . . . . . . . 9 |- ( ( F : V --> B /\ ( p .x. q ) e. V ) -> ( F ` ( p .x. q ) ) e. B )
17	8, 15, 16	syl2an2r 603	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( F ` ( p .x. q ) ) e. B )
18	14, 17	opelxpd 4807	. . . . . . 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 3860	. . . . . 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 404	. . . . 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 4058	. . . 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 4058	. . 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 3284	. 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
24		dff2 5852	. 2 |- ( .xb : ( B X. B ) --> B <-> ( .xb Fn ( B X. B ) /\ .xb C_ ( ( B X. B ) X. B ) ) )
25	6, 23, 24	sylanbrc 421	1 |- ( ph -> .xb : ( B X. B ) --> B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   {csn 3709   <.cop 3712   U_ciun 4012   X. cxp 4772   Fn wfn 5372   -->wf 5373   -onto->wfo 5375   ` cfv 5377  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088

This theorem is used by:  imasaddf  13640  imasmulf  13643  qusaddflemg  13655