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Theorem fnlimfv 39296
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 2761 . . . . 5 𝑦𝐷
4 nfcv 2761 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
5 nfcv 2761 . . . . . 6 𝑥
6 nfcv 2761 . . . . . . 7 𝑥𝑍
7 fnlimfv.2 . . . . . . . . 9 𝑥𝐹
8 nfcv 2761 . . . . . . . . 9 𝑥𝑚
97, 8nffv 6155 . . . . . . . 8 𝑥(𝐹𝑚)
10 nfcv 2761 . . . . . . . 8 𝑥𝑦
119, 10nffv 6155 . . . . . . 7 𝑥((𝐹𝑚)‘𝑦)
126, 11nfmpt 4706 . . . . . 6 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
135, 12nffv 6155 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
14 fveq2 6148 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
1514mpteq2dv 4705 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
1615fveq2d 6152 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
172, 3, 4, 13, 16cbvmptf 4708 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
181, 17eqtri 2643 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
1918a1i 11 . 2 (𝜑𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))))
20 fveq2 6148 . . . . 5 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
2120mpteq2dv 4705 . . . 4 (𝑦 = 𝑋 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
2221fveq2d 6152 . . 3 (𝑦 = 𝑋 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
2322adantl 482 . 2 ((𝜑𝑦 = 𝑋) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
24 fnlimfv.4 . 2 (𝜑𝑋𝐷)
25 fvex 6158 . . 3 ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V
2625a1i 11 . 2 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ V)
2719, 23, 24, 26fvmptd 6245 1 (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wnfc 2748  Vcvv 3186  cmpt 4673  cfv 5847  cli 14149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  fnlimcnv  39300  smflimlem2  40284
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