Theorem ofc12 5995

Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
ofc12.1	|- ( ph -> A e. V )
ofc12.2	|- ( ph -> B e. W )
ofc12.3	|- ( ph -> C e. X )

Assertion

Ref	Expression
ofc12	|- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Proof of Theorem ofc12

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofc12.1	. . 3 |- ( ph -> A e. V )
2		ofc12.2	. . . 4 |- ( ph -> B e. W )
3	2	adantr 274	. . 3 |- ( ( ph /\ x e. A ) -> B e. W )
4		ofc12.3	. . . 4 |- ( ph -> C e. X )
5	4	adantr 274	. . 3 |- ( ( ph /\ x e. A ) -> C e. X )
6		fconstmpt 4581	. . . 4 |- ( A X. { B } ) = ( x e. A |-> B )
7	6	a1i 9	. . 3 |- ( ph -> ( A X. { B } ) = ( x e. A |-> B ) )
8		fconstmpt 4581	. . . 4 |- ( A X. { C } ) = ( x e. A |-> C )
9	8	a1i 9	. . 3 |- ( ph -> ( A X. { C } ) = ( x e. A |-> C ) )
10	1, 3, 5, 7, 9	offval2 5990	. 2 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( x e. A |-> ( B R C ) ) )
11		fconstmpt 4581	. 2 |- ( A X. { ( B R C ) } ) = ( x e. A |-> ( B R C ) )
12	10, 11	syl6eqr 2188	1 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   {csn 3522   |-> cmpt 3984   X. cxp 4532  (class class class)co 5767   oFcof 5973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-of 5975

This theorem is referenced by: (None)