Theorem invffval 16418

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
invfval.b	|- B = ( Base ` C )
invfval.n	|- N = (Inv ` C )
invfval.c	|- ( ph -> C e. Cat )
invfval.x	|- ( ph -> X e. B )
invfval.y	|- ( ph -> Y e. B )
invfval.s	|- S = (Sect ` C )

Assertion

Ref	Expression
invffval	|- ( ph -> N = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )

Distinct variable groups:   x, y, B   ph, x, y   x, X, y   x, Y, y   x, C, y   x, S, y

Allowed substitution hints:   N( x, y)

Proof of Theorem invffval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.n	. 2 |- N = (Inv ` C )
2		invfval.c	. . 3 |- ( ph -> C e. Cat )
3		fveq2 6191	. . . . . 6 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
4		invfval.b	. . . . . 6 |- B = ( Base ` C )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( c = C -> ( Base ` c ) = B )
6		fveq2 6191	. . . . . . . 8 |- ( c = C -> (Sect ` c ) = (Sect ` C ) )
7		invfval.s	. . . . . . . 8 |- S = (Sect ` C )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( c = C -> (Sect ` c ) = S )
9	8	oveqd 6667	. . . . . 6 |- ( c = C -> ( x (Sect ` c ) y ) = ( x S y ) )
10	8	oveqd 6667	. . . . . . 7 |- ( c = C -> ( y (Sect ` c ) x ) = ( y S x ) )
11	10	cnveqd 5298	. . . . . 6 |- ( c = C -> `' ( y (Sect ` c ) x ) = `' ( y S x ) )
12	9, 11	ineq12d 3815	. . . . 5 |- ( c = C -> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) = ( ( x S y ) i^i `' ( y S x ) ) )
13	5, 5, 12	mpt2eq123dv 6717	. . . 4 |- ( c = C -> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )
14		df-inv 16408	. . . 4 |- Inv = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) )
15		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
16	4, 15	eqeltri 2697	. . . . 5 |- B e. _V
17	16, 16	mpt2ex 7247	. . . 4 |- ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) e. _V
18	13, 14, 17	fvmpt 6282	. . 3 |- ( C e. Cat -> (Inv ` C ) = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )
19	2, 18	syl 17	. 2 |- ( ph -> (Inv ` C ) = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )
20	1, 19	syl5eq 2668	1 |- ( ph -> N = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   `'ccnv 5113   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Catccat 16325  Sectcsect 16404  Invcinv 16405

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-rep 4771  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-reu 2919  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-pw 4160  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-inv 16408

This theorem is referenced by:  invfval  16419  isoval  16425