Theorem stdbdmetval 22319

Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypothesis

Ref	Expression
stdbdmet.1	|- D = ( x e. X , y e. X |-> if ( ( x C y ) <_ R , ( x C y ) , R ) )

Assertion

Ref	Expression
stdbdmetval	|- ( ( R e. V /\ A e. X /\ B e. X ) -> ( A D B ) = if ( ( A C B ) <_ R , ( A C B ) , R ) )

Distinct variable groups:   x, y, A   x, C, y   x, B, y   x, R, y   x, X, y

Allowed substitution hints:   D( x, y)   V( x, y)

Proof of Theorem stdbdmetval

Step	Hyp	Ref	Expression
1		ovex 6678	. . . 4 |- ( A C B ) e. _V
2		ifexg 4157	. . . 4 |- ( ( ( A C B ) e. _V /\ R e. V ) -> if ( ( A C B ) <_ R , ( A C B ) , R ) e. _V )
3	1, 2	mpan 706	. . 3 |- ( R e. V -> if ( ( A C B ) <_ R , ( A C B ) , R ) e. _V )
4		oveq12 6659	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( x C y ) = ( A C B ) )
5	4	breq1d 4663	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( x C y ) <_ R <-> ( A C B ) <_ R ) )
6	5, 4	ifbieq1d 4109	. . . 4 |- ( ( x = A /\ y = B ) -> if ( ( x C y ) <_ R , ( x C y ) , R ) = if ( ( A C B ) <_ R , ( A C B ) , R ) )
7		stdbdmet.1	. . . 4 |- D = ( x e. X , y e. X |-> if ( ( x C y ) <_ R , ( x C y ) , R ) )
8	6, 7	ovmpt2ga 6790	. . 3 |- ( ( A e. X /\ B e. X /\ if ( ( A C B ) <_ R , ( A C B ) , R ) e. _V ) -> ( A D B ) = if ( ( A C B ) <_ R , ( A C B ) , R ) )
9	3, 8	syl3an3 1361	. 2 |- ( ( A e. X /\ B e. X /\ R e. V ) -> ( A D B ) = if ( ( A C B ) <_ R , ( A C B ) , R ) )
10	9	3comr 1273	1 |- ( ( R e. V /\ A e. X /\ B e. X ) -> ( A D B ) = if ( ( A C B ) <_ R , ( A C B ) , R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652   <_ cle 10075

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  stdbdbl  22322