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Theorem fnlimfv 42671
 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 . . 3 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
2 fnlimfv.1 . . . 4 𝑥𝐷
3 nfcv 2919 . . . 4 𝑦𝐷
4 nfcv 2919 . . . 4 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
5 nfcv 2919 . . . . 5 𝑥
6 nfcv 2919 . . . . . 6 𝑥𝑍
7 fnlimfv.2 . . . . . . . 8 𝑥𝐹
8 nfcv 2919 . . . . . . . 8 𝑥𝑚
97, 8nffv 6668 . . . . . . 7 𝑥(𝐹𝑚)
10 nfcv 2919 . . . . . . 7 𝑥𝑦
119, 10nffv 6668 . . . . . 6 𝑥((𝐹𝑚)‘𝑦)
126, 11nfmpt 5129 . . . . 5 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
135, 12nffv 6668 . . . 4 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
14 fveq2 6658 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
1514mpteq2dv 5128 . . . . 5 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
1615fveq2d 6662 . . . 4 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
172, 3, 4, 13, 16cbvmptf 5131 . . 3 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
181, 17eqtri 2781 . 2 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
19 fveq2 6658 . . . 4 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
2019mpteq2dv 5128 . . 3 (𝑦 = 𝑋 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
2120fveq2d 6662 . 2 (𝑦 = 𝑋 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
22 fnlimfv.4 . 2 (𝜑𝑋𝐷)
23 fvexd 6673 . 2 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V)
2418, 21, 22, 23fvmptd3 6782 1 (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2899  Vcvv 3409   ↦ cmpt 5112  ‘cfv 6335   ⇝ cli 14889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343 This theorem is referenced by:  fnlimcnv  42675  smflimlem2  43771
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