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Theorem imasaddflem 16190
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 ) ) >. } )
imasaddflem.c |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
Assertion
Ref Expression
imasaddflem |- ( 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)   .x. ( a, b)
Proof of Theorem imasaddflem
StepHypRef 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 ) ) >. } )
41, 2, 3imasaddfnlem 16188 . 2 |- ( ph -> .xb Fn ( B X. B ) )
5 fof 6115 . . . . . . . . . . . 12 |- ( F : V -onto-> B -> F : V --> B )
61, 5syl 17 . . . . . . . . . . 11 |- ( ph -> F : V --> B )
7 ffvelrn 6357 . . . . . . . . . . . . 13 |- ( ( F : V --> B /\ p e. V ) -> ( F ` p ) e. B )
8 ffvelrn 6357 . . . . . . . . . . . . 13 |- ( ( F : V --> B /\ q e. V ) -> ( F ` q ) e. B )
97, 8anim12dan 882 . . . . . . . . . . . 12 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> ( ( F ` p ) e. B /\ ( F ` q ) e. B ) )
10 opelxpi 5148 . . . . . . . . . . . 12 |- ( ( ( F ` p ) e. B /\ ( F ` q ) e. B ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
119, 10syl 17 . . . . . . . . . . 11 |- ( ( F : V --> B /\ ( p e. V /\ q e. V ) ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
126, 11sylan 488 . . . . . . . . . 10 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) )
13 imasaddflem.c . . . . . . . . . . 11 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
14 ffvelrn 6357 . . . . . . . . . . . 12 |- ( ( F : V --> B /\ ( p .x. q ) e. V ) -> ( F ` ( p .x. q ) ) e. B )
156, 14sylan 488 . . . . . . . . . . 11 |- ( ( ph /\ ( p .x. q ) e. V ) -> ( F ` ( p .x. q ) ) e. B )
1613, 15syldan 487 . . . . . . . . . 10 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( F ` ( p .x. q ) ) e. B )
17 opelxpi 5148 . . . . . . . . . 10 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. ( B X. B ) /\ ( F ` ( p .x. q ) ) e. B ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. ( ( B X. B ) X. B ) )
1812, 16, 17syl2anc 693 . . . . . . . . 9 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. ( ( B X. B ) X. B ) )
1918snssd 4340 . . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
2019anassrs 680 . . . . . . 7 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
2120ralrimiva 2966 . . . . . 6 |- ( ( ph /\ p e. V ) -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
22 iunss 4561 . . . . . 6 |- ( U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) <-> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
2321, 22sylibr 224 . . . . 5 |- ( ( ph /\ p e. V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
2423ralrimiva 2966 . . . 4 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
25 iunss 4561 . . . 4 |- ( U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) <-> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
2624, 25sylibr 224 . . 3 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } C_ ( ( B X. B ) X. B ) )
273, 26eqsstrd 3639 . 2 |- ( ph -> .xb C_ ( ( B X. B ) X. B ) )
28 dff2 6371 . 2 |- ( .xb : ( B X. B ) --> B <-> ( .xb Fn ( B X. B ) /\ .xb C_ ( ( B X. B ) X. B ) ) )
294, 27, 28sylanbrc 698 1 |- ( ph -> .xb : ( B X. B ) --> B )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896
This theorem is referenced by:  imasaddf  16193  imasmulf  16196  qusaddflem  16212
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