Theorem ofc12 6182

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 276	. . 3 |- ( ( ph /\ x e. A ) -> B e. W )
4		ofc12.3	. . . 4 |- ( ph -> C e. X )
5	4	adantr 276	. . 3 |- ( ( ph /\ x e. A ) -> C e. X )
6		fconstmpt 4722	. . . 4 |- ( A X. { B } ) = ( x e. A |-> B )
7	6	a1i 9	. . 3 |- ( ph -> ( A X. { B } ) = ( x e. A |-> B ) )
8		fconstmpt 4722	. . . 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 6174	. 2 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( x e. A |-> ( B R C ) ) )
11		fconstmpt 4722	. 2 |- ( A X. { ( B R C ) } ) = ( x e. A |-> ( B R C ) )
12	10, 11	eqtr4di 2256	1 |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {csn 3633   |-> cmpt 4105   X. cxp 4673  (class class class)co 5944   oFcof 6156

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-in1 615  ax-in2 616  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-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  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-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  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  df-oprab 5948  df-mpo 5949  df-of 6158

This theorem is referenced by: (None)