Theorem fnlimfv 39895

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	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
2		fnlimfv.1	. . . . 5 ⊢ Ⅎ𝑥𝐷
3		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦𝐷
4		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
5		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥 ⇝
6		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥𝑍
7		fnlimfv.2	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
8		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑚
9	7, 8	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑚)
10		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 6198	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑦)
12	6, 11	nfmpt 4746	. . . . . 6 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
13	5, 12	nffv 6198	. . . . 5 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
14		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
15	14	mpteq2dv 4745	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
16	15	fveq2d 6195	. . . . 5 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
17	2, 3, 4, 13, 16	cbvmptf 4748	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
18	1, 17	eqtri 2644	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
19	18	a1i 11	. 2 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))))
20		fveq2 6191	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
21	20	mpteq2dv 4745	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
22	21	fveq2d 6195	. . 3 ⊢ (𝑦 = 𝑋 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
23	22	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑋) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
24		fnlimfv.4	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
25		fvexd 6203	. 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V)
26	19, 23, 24, 25	fvmptd 6288	1 ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  Vcvv 3200   ↦ cmpt 4729  ‘cfv 5888   ⇝ cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fnlimcnv  39899  smflimlem2  40980