Theorem aovmpt4g 41281

Description: Value of a function given by the "maps to" notation, analogous to ovmpt4g 6783. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aovmpt4g.3	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
aovmpt4g	|- ( ( x e. A /\ y e. B /\ C e. V ) -> (( x F y)) = C )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, V, y

Allowed substitution hints:   F( x, y)

Proof of Theorem aovmpt4g

Step	Hyp	Ref	Expression
1		aovmpt4g.3	. . . . . . 7 |- F = ( x e. A , y e. B |-> C )
2	1	dmmpt2g 7243	. . . . . 6 |- ( C e. V -> dom F = ( A X. B ) )
3		opelxpi 5148	. . . . . . 7 |- ( ( x e. A /\ y e. B ) -> <. x , y >. e. ( A X. B ) )
4		eleq2 2690	. . . . . . 7 |- ( dom F = ( A X. B ) -> ( <. x , y >. e. dom F <-> <. x , y >. e. ( A X. B ) ) )
5	3, 4	syl5ibr 236	. . . . . 6 |- ( dom F = ( A X. B ) -> ( ( x e. A /\ y e. B ) -> <. x , y >. e. dom F ) )
6	2, 5	syl 17	. . . . 5 |- ( C e. V -> ( ( x e. A /\ y e. B ) -> <. x , y >. e. dom F ) )
7	6	impcom 446	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ C e. V ) -> <. x , y >. e. dom F )
8	7	3impa 1259	. . 3 |- ( ( x e. A /\ y e. B /\ C e. V ) -> <. x , y >. e. dom F )
9	1	mpt2fun 6762	. . . 4 |- Fun F
10		funres 5929	. . . 4 |- ( Fun F -> Fun ( F |` { <. x , y >. } ) )
11	9, 10	ax-mp 5	. . 3 |- Fun ( F |` { <. x , y >. } )
12		df-dfat 41196	. . . 4 |- ( F defAt <. x , y >. <-> ( <. x , y >. e. dom F /\ Fun ( F |` { <. x , y >. } ) ) )
13		aovfundmoveq 41261	. . . 4 |- ( F defAt <. x , y >. -> (( x F y)) = ( x F y ) )
14	12, 13	sylbir 225	. . 3 |- ( ( <. x , y >. e. dom F /\ Fun ( F |` { <. x , y >. } ) ) -> (( x F y)) = ( x F y ) )
15	8, 11, 14	sylancl 694	. 2 |- ( ( x e. A /\ y e. B /\ C e. V ) -> (( x F y)) = ( x F y ) )
16	1	ovmpt4g 6783	. 2 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )
17	15, 16	eqtrd 2656	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> (( x F y)) = C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   X. cxp 5112   dom cdm 5114   |` cres 5116   Fun wfun 5882  (class class class)co 6650   |-> cmpt2 6652   defAt wdfat 41193   ((caov 41195

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)