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Theorem fnlimfv 38730
 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 2751 . . . . 5 𝑦𝐷
4 nfcv 2751 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
5 nfcv 2751 . . . . . 6 𝑥
6 nfcv 2751 . . . . . . 7 𝑥𝑍
7 fnlimfv.2 . . . . . . . . 9 𝑥𝐹
8 nfcv 2751 . . . . . . . . 9 𝑥𝑚
97, 8nffv 6110 . . . . . . . 8 𝑥(𝐹𝑚)
10 nfcv 2751 . . . . . . . 8 𝑥𝑦
119, 10nffv 6110 . . . . . . 7 𝑥((𝐹𝑚)‘𝑦)
126, 11nfmpt 4674 . . . . . 6 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
135, 12nffv 6110 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
14 fveq2 6103 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
1514mpteq2dv 4673 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
1615fveq2d 6107 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
172, 3, 4, 13, 16cbvmptf 4676 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
181, 17eqtri 2632 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
1918a1i 11 . 2 (𝜑𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))))
20 fveq2 6103 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
2120mpteq2dv 4673 . . . 4 (𝑦 = 𝑋 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
2221fveq2d 6107 . . 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 fvex 6113 . . 3 ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V
2625a1i 11 . 2 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V)
2719, 23, 24, 26fvmptd 6197 1 (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804   ⇝ cli 14063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  fnlimcnv  38734  smflimlem2  39658
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