Theorem evlf2 16858

Description: Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlfval.e	|- E = ( C evalF D )
evlfval.c	|- ( ph -> C e. Cat )
evlfval.d	|- ( ph -> D e. Cat )
evlfval.b	|- B = ( Base ` C )
evlfval.h	|- H = ( Hom ` C )
evlfval.o	|- .x. = (comp ` D )
evlfval.n	|- N = ( C Nat D )
evlf2.f	|- ( ph -> F e. ( C Func D ) )
evlf2.g	|- ( ph -> G e. ( C Func D ) )
evlf2.x	|- ( ph -> X e. B )
evlf2.y	|- ( ph -> Y e. B )
evlf2.l	|- L = ( <. F , X >. ( 2nd ` E ) <. G , Y >. )

Assertion

Ref	Expression
evlf2	|- ( ph -> L = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )

Distinct variable groups:   g, a, C   D, a, g   g, H   F, a, g   N, a, g   G, a, g   ph, a, g   .x. , a, g   X, a, g   Y, a, g

Allowed substitution hints:   B( g, a)   E( g, a)   H( a)   L( g, a)

Proof of Theorem evlf2

Dummy variables f m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlf2.l	. 2 |- L = ( <. F , X >. ( 2nd ` E ) <. G , Y >. )
2		evlfval.e	. . . . 5 |- E = ( C evalF D )
3		evlfval.c	. . . . 5 |- ( ph -> C e. Cat )
4		evlfval.d	. . . . 5 |- ( ph -> D e. Cat )
5		evlfval.b	. . . . 5 |- B = ( Base ` C )
6		evlfval.h	. . . . 5 |- H = ( Hom ` C )
7		evlfval.o	. . . . 5 |- .x. = (comp ` D )
8		evlfval.n	. . . . 5 |- N = ( C Nat D )
9	2, 3, 4, 5, 6, 7, 8	evlfval 16857	. . . 4 |- ( ph -> E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
10		ovex 6678	. . . . . 6 |- ( C Func D ) e. _V
11		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
12	5, 11	eqeltri 2697	. . . . . 6 |- B e. _V
13	10, 12	mpt2ex 7247	. . . . 5 |- ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) e. _V
14	10, 12	xpex 6962	. . . . . 6 |- ( ( C Func D ) X. B ) e. _V
15	14, 14	mpt2ex 7247	. . . . 5 |- ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) e. _V
16	13, 15	op2ndd 7179	. . . 4 |- ( E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
17	9, 16	syl 17	. . 3 |- ( ph -> ( 2nd ` E ) = ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) )
18		fvexd 6203	. . . 4 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> ( 1st ` x ) e. _V )
19		simprl 794	. . . . . 6 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> x = <. F , X >. )
20	19	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> ( 1st ` x ) = ( 1st ` <. F , X >. ) )
21		evlf2.f	. . . . . . 7 |- ( ph -> F e. ( C Func D ) )
22		evlf2.x	. . . . . . 7 |- ( ph -> X e. B )
23		op1stg 7180	. . . . . . 7 |- ( ( F e. ( C Func D ) /\ X e. B ) -> ( 1st ` <. F , X >. ) = F )
24	21, 22, 23	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. F , X >. ) = F )
25	24	adantr 481	. . . . 5 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> ( 1st ` <. F , X >. ) = F )
26	20, 25	eqtrd 2656	. . . 4 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> ( 1st ` x ) = F )
27		fvexd 6203	. . . . 5 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> ( 1st ` y ) e. _V )
28		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> y = <. G , Y >. )
29	28	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> ( 1st ` y ) = ( 1st ` <. G , Y >. ) )
30		evlf2.g	. . . . . . . 8 |- ( ph -> G e. ( C Func D ) )
31		evlf2.y	. . . . . . . 8 |- ( ph -> Y e. B )
32		op1stg 7180	. . . . . . . 8 |- ( ( G e. ( C Func D ) /\ Y e. B ) -> ( 1st ` <. G , Y >. ) = G )
33	30, 31, 32	syl2anc 693	. . . . . . 7 |- ( ph -> ( 1st ` <. G , Y >. ) = G )
34	33	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> ( 1st ` <. G , Y >. ) = G )
35	29, 34	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> ( 1st ` y ) = G )
36		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> m = F )
37		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> n = G )
38	36, 37	oveq12d 6668	. . . . . 6 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( m N n ) = ( F N G ) )
39	19	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> x = <. F , X >. )
40	39	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` x ) = ( 2nd ` <. F , X >. ) )
41		op2ndg 7181	. . . . . . . . . 10 |- ( ( F e. ( C Func D ) /\ X e. B ) -> ( 2nd ` <. F , X >. ) = X )
42	21, 22, 41	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 2nd ` <. F , X >. ) = X )
43	42	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` <. F , X >. ) = X )
44	40, 43	eqtrd 2656	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` x ) = X )
45	28	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> y = <. G , Y >. )
46	45	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` y ) = ( 2nd ` <. G , Y >. ) )
47		op2ndg 7181	. . . . . . . . . 10 |- ( ( G e. ( C Func D ) /\ Y e. B ) -> ( 2nd ` <. G , Y >. ) = Y )
48	30, 31, 47	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 2nd ` <. G , Y >. ) = Y )
49	48	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` <. G , Y >. ) = Y )
50	46, 49	eqtrd 2656	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` y ) = Y )
51	44, 50	oveq12d 6668	. . . . . 6 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( 2nd ` x ) H ( 2nd ` y ) ) = ( X H Y ) )
52	36	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 1st ` m ) = ( 1st ` F ) )
53	52, 44	fveq12d 6197	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( 1st ` m ) ` ( 2nd ` x ) ) = ( ( 1st ` F ) ` X ) )
54	52, 50	fveq12d 6197	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( 1st ` m ) ` ( 2nd ` y ) ) = ( ( 1st ` F ) ` Y ) )
55	53, 54	opeq12d 4410	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. = <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. )
56	37	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 1st ` n ) = ( 1st ` G ) )
57	56, 50	fveq12d 6197	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( 1st ` n ) ` ( 2nd ` y ) ) = ( ( 1st ` G ) ` Y ) )
58	55, 57	oveq12d 6668	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) = ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) )
59	50	fveq2d 6195	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( a ` ( 2nd ` y ) ) = ( a ` Y ) )
60	36	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( 2nd ` m ) = ( 2nd ` F ) )
61	60, 44, 50	oveq123d 6671	. . . . . . . 8 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) = ( X ( 2nd ` F ) Y ) )
62	61	fveq1d 6193	. . . . . . 7 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) = ( ( X ( 2nd ` F ) Y ) ` g ) )
63	58, 59, 62	oveq123d 6671	. . . . . 6 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) = ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) )
64	38, 51, 63	mpt2eq123dv 6717	. . . . 5 |- ( ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) /\ n = G ) -> ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )
65	27, 35, 64	csbied2 3561	. . . 4 |- ( ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) /\ m = F ) -> [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )
66	18, 26, 65	csbied2 3561	. . 3 |- ( ( ph /\ ( x = <. F , X >. /\ y = <. G , Y >. ) ) -> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m N n ) , g e. ( ( 2nd ` x ) H ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. .x. ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )
67		opelxpi 5148	. . . 4 |- ( ( F e. ( C Func D ) /\ X e. B ) -> <. F , X >. e. ( ( C Func D ) X. B ) )
68	21, 22, 67	syl2anc 693	. . 3 |- ( ph -> <. F , X >. e. ( ( C Func D ) X. B ) )
69		opelxpi 5148	. . . 4 |- ( ( G e. ( C Func D ) /\ Y e. B ) -> <. G , Y >. e. ( ( C Func D ) X. B ) )
70	30, 31, 69	syl2anc 693	. . 3 |- ( ph -> <. G , Y >. e. ( ( C Func D ) X. B ) )
71		ovex 6678	. . . . 5 |- ( F N G ) e. _V
72		ovex 6678	. . . . 5 |- ( X H Y ) e. _V
73	71, 72	mpt2ex 7247	. . . 4 |- ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) e. _V
74	73	a1i 11	. . 3 |- ( ph -> ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) e. _V )
75	17, 66, 68, 70, 74	ovmpt2d 6788	. 2 |- ( ph -> ( <. F , X >. ( 2nd ` E ) <. G , Y >. ) = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )
76	1, 75	syl5eq 2668	1 |- ( ph -> L = ( a e. ( F N G ) , g e. ( X H Y ) |-> ( ( a ` Y ) ( <. ( ( 1st ` F ) ` X ) , ( ( 1st ` F ) ` Y ) >. .x. ( ( 1st ` G ) ` Y ) ) ( ( X ( 2nd ` F ) Y ) ` g ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   eval_F cevlf 16849

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-evlf 16853

This theorem is referenced by:  evlf2val  16859  evlfcl  16862