Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnlimfv Structured version   Visualization version   GIF version

Theorem fnlimfv 40213
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 𝑥𝐷
fnlimfv.2 𝑥𝐹
fnlimfv.3 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
fnlimfv.4 (𝜑𝑋𝐷)
Assertion
Ref Expression
fnlimfv (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
Distinct variable groups:   𝑚,𝑋   𝑥,𝑍   𝑥,𝑚
Allowed substitution hints:   𝜑(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.
StepHypRef Expression
1 fnlimfv.3 . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
2 fnlimfv.1 . . . . 5 𝑥𝐷
3 nfcv 2793 . . . . 5 𝑦𝐷
4 nfcv 2793 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
5 nfcv 2793 . . . . . 6 𝑥
6 nfcv 2793 . . . . . . 7 𝑥𝑍
7 fnlimfv.2 . . . . . . . . 9 𝑥𝐹
8 nfcv 2793 . . . . . . . . 9 𝑥𝑚
97, 8nffv 6236 . . . . . . . 8 𝑥(𝐹𝑚)
10 nfcv 2793 . . . . . . . 8 𝑥𝑦
119, 10nffv 6236 . . . . . . 7 𝑥((𝐹𝑚)‘𝑦)
126, 11nfmpt 4779 . . . . . 6 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
135, 12nffv 6236 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
14 fveq2 6229 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
1514mpteq2dv 4778 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
1615fveq2d 6233 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
172, 3, 4, 13, 16cbvmptf 4781 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
181, 17eqtri 2673 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
1918a1i 11 . 2 (𝜑𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))))
20 fveq2 6229 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
2120mpteq2dv 4778 . . . 4 (𝑦 = 𝑋 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
2221fveq2d 6233 . . 3 (𝑦 = 𝑋 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
2322adantl 481 . 2 ((𝜑𝑦 = 𝑋) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
24 fnlimfv.4 . 2 (𝜑𝑋𝐷)
25 fvexd 6241 . 2 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V)
2619, 23, 24, 25fvmptd 6327 1 (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wnfc 2780  Vcvv 3231  cmpt 4762  cfv 5926  cli 14259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  fnlimcnv  40217  smflimlem2  41301
  Copyright terms: Public domain W3C validator