Theorem imasaddflemg 13148

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 13146	. 2 |- ( ph -> .xb Fn ( B X. B ) )
7		fof 5498	. . . . . . . . . 10 |- ( F : V -onto-> B -> F : V --> B )
8	1, 7	syl 14	. . . . . . . . 9 |- ( ph -> F : V --> B )
9		ffvelcdm 5713	. . . . . . . . . . 11 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
10		ffvelcdm 5713	. . . . . . . . . . 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 4707	. . . . . . . . . 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 5713	. . . . . . . . 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 4708	. . . . . . 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 3778	. . . . . 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 3973	. . . 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 3973	. . 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 3229	. 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
24		dff2 5724	. 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 981   = wceq 1373   e. wcel 2176   C_ wss 3166   {csn 3633   <.cop 3636   U_ciun 3927   X. cxp 4673   Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947

This theorem is referenced by:  imasaddf  13151  imasmulf  13154  qusaddflemg  13166