Theorem fnlimfv 39895

Description: The value of the limit function G at any point of its domain D. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
fnlimfv.1	|- F/_ x D
fnlimfv.2	|- F/_ x F
fnlimfv.3	|- G = ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )
fnlimfv.4	|- ( ph -> X e. D )

Assertion

Ref	Expression
fnlimfv	|- ( ph -> ( G ` X ) = ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) )

Distinct variable groups:   m, X   x, Z   x, m

Allowed substitution hints:   ph( x, m)   D( x, m)   F( x, m)   G( x, m)   X( x)   Z( m)

Proof of Theorem fnlimfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnlimfv.3	. . . 4 |- G = ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )
2		fnlimfv.1	. . . . 5 |- F/_ x D
3		nfcv 2764	. . . . 5 |- F/_ y D
4		nfcv 2764	. . . . 5 |- F/_ y ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) )
5		nfcv 2764	. . . . . 6 |- F/_ x ~~>
6		nfcv 2764	. . . . . . 7 |- F/_ x Z
7		fnlimfv.2	. . . . . . . . 9 |- F/_ x F
8		nfcv 2764	. . . . . . . . 9 |- F/_ x m
9	7, 8	nffv 6198	. . . . . . . 8 |- F/_ x ( F ` m )
10		nfcv 2764	. . . . . . . 8 |- F/_ x y
11	9, 10	nffv 6198	. . . . . . 7 |- F/_ x ( ( F ` m ) ` y )
12	6, 11	nfmpt 4746	. . . . . 6 |- F/_ x ( m e. Z |-> ( ( F ` m ) ` y ) )
13	5, 12	nffv 6198	. . . . 5 |- F/_ x ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) )
14		fveq2 6191	. . . . . . 7 |- ( x = y -> ( ( F ` m ) ` x ) = ( ( F ` m ) ` y ) )
15	14	mpteq2dv 4745	. . . . . 6 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( m e. Z |-> ( ( F ` m ) ` y ) ) )
16	15	fveq2d 6195	. . . . 5 |- ( x = y -> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) = ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) )
17	2, 3, 4, 13, 16	cbvmptf 4748	. . . 4 |- ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) ) = ( y e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) )
18	1, 17	eqtri 2644	. . 3 |- G = ( y e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) )
19	18	a1i 11	. 2 |- ( ph -> G = ( y e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) ) )
20		fveq2 6191	. . . . 5 |- ( y = X -> ( ( F ` m ) ` y ) = ( ( F ` m ) ` X ) )
21	20	mpteq2dv 4745	. . . 4 |- ( y = X -> ( m e. Z |-> ( ( F ` m ) ` y ) ) = ( m e. Z |-> ( ( F ` m ) ` X ) ) )
22	21	fveq2d 6195	. . 3 |- ( y = X -> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) = ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) )
23	22	adantl 482	. 2 |- ( ( ph /\ y = X ) -> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` y ) ) ) = ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) )
24		fnlimfv.4	. 2 |- ( ph -> X e. D )
25		fvexd 6203	. 2 |- ( ph -> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) e. _V )
26	19, 23, 24, 25	fvmptd 6288	1 |- ( ph -> ( G ` X ) = ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888   ~~> cli 14215

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fnlimcnv  39899  smflimlem2  40980