Theorem imasaddflemg 12959

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 12957	. 2 |- ( ph -> .xb Fn ( B X. B ) )
7		fof 5480	. . . . . . . . . 10 |- ( F : V -onto-> B -> F : V --> B )
8	1, 7	syl 14	. . . . . . . . 9 |- ( ph -> F : V --> B )
9		ffvelcdm 5695	. . . . . . . . . . 11 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
10		ffvelcdm 5695	. . . . . . . . . . 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 4695	. . . . . . . . . 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 5695	. . . . . . . . 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 4696	. . . . . . 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 3767	. . . . . 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 3962	. . . 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 3962	. . 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 3219	. 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
24		dff2 5706	. 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 980   = wceq 1364   e. wcel 2167   C_ wss 3157   {csn 3622   <.cop 3625   U_ciun 3916   X. cxp 4661   Fn wfn 5253   -->wf 5254   -onto->wfo 5256   ` cfv 5258  (class class class)co 5922

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925

This theorem is referenced by:  imasaddf  12962  imasmulf  12965  qusaddflemg  12977