Theorem fnlimfv 45574

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	⊢ Ⅎ𝑥𝐷
fnlimfv.2	⊢ Ⅎ𝑥𝐹
fnlimfv.3	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
fnlimfv.4	⊢ (𝜑 → 𝑋 ∈ 𝐷)

Assertion

Ref	Expression
fnlimfv	⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘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.

Step	Hyp	Ref	Expression
1		fnlimfv.3	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
2		fnlimfv.1	. . . 4 ⊢ Ⅎ𝑥𝐷
3		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦𝐷
4		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
5		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥 ⇝
6		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥𝑍
7		fnlimfv.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
8		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
9	7, 8	nffv 6925	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
10		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 6925	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑦)
12	6, 11	nfmpt 5273	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
13	5, 12	nffv 6925	. . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
14		fveq2 6915	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
15	14	mpteq2dv 5268	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
16	15	fveq2d 6919	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
17	2, 3, 4, 13, 16	cbvmptf 5275	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
18	1, 17	eqtri 2768	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
19		fveq2 6915	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
20	19	mpteq2dv 5268	. . 3 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
21	20	fveq2d 6919	. 2 ⊢ (𝑦 = 𝑋 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
22		fnlimfv.4	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
23		fvexd 6930	. 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V)
24	18, 21, 22, 23	fvmptd3 7047	1 ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488   ↦ cmpt 5249  ‘cfv 6568   ⇝ cli 15524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-iota 6520  df-fun 6570  df-fv 6576

This theorem is referenced by:  fnlimcnv  45578  smflimlem2  46683