Theorem fnlimfv 41951

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 2979	. . . 4 ⊢ Ⅎ𝑦𝐷
4		nfcv 2979	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
5		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥 ⇝
6		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑥𝑍
7		fnlimfv.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
8		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
9	7, 8	nffv 6682	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
10		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
11	9, 10	nffv 6682	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑦)
12	6, 11	nfmpt 5165	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
13	5, 12	nffv 6682	. . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
14		fveq2 6672	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
15	14	mpteq2dv 5164	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
16	15	fveq2d 6676	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
17	2, 3, 4, 13, 16	cbvmptf 5167	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
18	1, 17	eqtri 2846	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
19		fveq2 6672	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
20	19	mpteq2dv 5164	. . 3 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
21	20	fveq2d 6676	. 2 ⊢ (𝑦 = 𝑋 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
22		fnlimfv.4	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
23		fvexd 6687	. 2 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ V)
24	18, 21, 22, 23	fvmptd3 6793	1 ⊢ (𝜑 → (𝐺‘𝑋) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2963  Vcvv 3496   ↦ cmpt 5148  ‘cfv 6357   ⇝ cli 14843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by:  fnlimcnv  41955  smflimlem2  43055