Theorem fnxpc 16816

Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)

Assertion

Ref	Expression
fnxpc	|- X.c Fn ( _V X. _V )

Proof of Theorem fnxpc

Dummy variables f b g h r s u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xpc 16812	. 2 |- X.c = ( r e. _V , s e. _V |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
2		tpex 6957	. . . 4 |- { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } e. _V
3	2	csbex 4793	. . 3 |- [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } e. _V
4	3	csbex 4793	. 2 |- [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } e. _V
5	1, 4	fnmpt2i 7239	1 |- X.c Fn ( _V X. _V )

Syntax hints:   _Vcvv 3200   [_csb 3533   {ctp 4181   <.cop 4183   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   X._c cxpc 16808

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-fal 1489  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-tp 4182  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-xpc 16812

This theorem is referenced by:  xpcbas  16818  xpchomfval  16819  xpccofval  16822