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Theorem ulmdvlem2 26617

Description: Lemma for ulmdv 26619. (Contributed by Mario Carneiro, 8-May-2015.)

**Hypotheses**
Ref	Expression
ulmdv.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
ulmdv.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
ulmdv.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulmdv.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑋))
ulmdv.g	⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
ulmdv.l	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
ulmdv.u	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)

**Assertion**
Ref	Expression
ulmdvlem2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)

Distinct variable groups: 𝑧,𝑘,𝐹 𝑧,𝐺 𝑧,𝐻 𝑘,𝑀 𝜑,𝑘,𝑧 𝑆,𝑘,𝑧 𝑘,𝑋,𝑧 𝑘,𝑍,𝑧 Allowed substitution hints: 𝐺(𝑘) 𝐻(𝑘) 𝑀(𝑧)

**Proof of Theorem ulmdvlem2**
Step	Hyp	Ref	Expression
1		ovex 7452	. . . . . 6 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V
2	1	rgenw 3085	. . . . 5 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V
3		eqid 2765	. . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))
4	3	fnmpt 6679	. . . . 5 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
5	2, 4	mp1i 14	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
6		ulmdv.u	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
7		ulmf2 26600	. . . 4 ⊢ (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍 ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_m 𝑋))
8	5, 6, 7	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_m 𝑋))
9	8	fvmptelcdm 7112	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑_m 𝑋))
10		elmapi 8852	. 2 ⊢ ((𝑆 D (𝐹‘𝑘)) ∈ (ℂ ↑_m 𝑋) → (𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ)
11		fdm 6719	. 2 ⊢ ((𝑆 D (𝐹‘𝑘)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
12	9, 10, 11	3syl 19	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 Vcvv 3457 {cpr 4593 class class class wbr 5111 ↦ cmpt 5194 dom cdm 5663 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ↑_m cmap 8830 ℂcc 11115 ℝcr 11116 ℤcz 12608 ℤ_≥cuz 12880 ⇝ cli 15561 D cdv 26075 ⇝_𝑢culm 26592

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-map 8832 df-pm 8833 df-neg 11461 df-z 12609 df-uz 12881 df-ulm 26593

This theorem is used by: ulmdvlem3 26618 ulmdv 26619

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