Theorem estrreslem2 18158

Description: Lemma 2 for estrres 18159. (Contributed by AV, 14-Mar-2020.)

Hypotheses

Ref	Expression
estrres.c	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
estrres.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
estrres.h	⊢ (𝜑 → 𝐻 ∈ 𝑋)
estrres.x	⊢ (𝜑 → · ∈ 𝑌)

Assertion

Ref	Expression
estrreslem2	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐶)

Proof of Theorem estrreslem2

Step	Hyp	Ref	Expression
1		eqidd 2735	. . . 4 ⊢ (𝜑 → (Base‘ndx) = (Base‘ndx))
2	1	3mix1d 1336	. . 3 ⊢ (𝜑 → ((Base‘ndx) = (Base‘ndx) ∨ (Base‘ndx) = (Hom ‘ndx) ∨ (Base‘ndx) = (comp‘ndx)))
3		fvex 6900	. . . 4 ⊢ (Base‘ndx) ∈ V
4		eltpg 4668	. . . 4 ⊢ ((Base‘ndx) ∈ V → ((Base‘ndx) ∈ {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)} ↔ ((Base‘ndx) = (Base‘ndx) ∨ (Base‘ndx) = (Hom ‘ndx) ∨ (Base‘ndx) = (comp‘ndx))))
5	3, 4	mp1i 13	. . 3 ⊢ (𝜑 → ((Base‘ndx) ∈ {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)} ↔ ((Base‘ndx) = (Base‘ndx) ∨ (Base‘ndx) = (Hom ‘ndx) ∨ (Base‘ndx) = (comp‘ndx))))
6	2, 5	mpbird 257	. 2 ⊢ (𝜑 → (Base‘ndx) ∈ {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)})
7		df-tp 4613	. . . . . 6 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} = ({⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ {⟨(comp‘ndx), · ⟩})
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} = ({⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ {⟨(comp‘ndx), · ⟩}))
9	8	dmeqd 5898	. . . 4 ⊢ (𝜑 → dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} = dom ({⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ {⟨(comp‘ndx), · ⟩}))
10		dmun 5903	. . . . 5 ⊢ dom ({⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ {⟨(comp‘ndx), · ⟩}) = (dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ dom {⟨(comp‘ndx), · ⟩})
11	10	a1i 11	. . . 4 ⊢ (𝜑 → dom ({⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ {⟨(comp‘ndx), · ⟩}) = (dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ dom {⟨(comp‘ndx), · ⟩}))
12		estrres.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
13		estrres.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
14		dmpropg 6217	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐻 ∈ 𝑋) → dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} = {(Base‘ndx), (Hom ‘ndx)})
15	12, 13, 14	syl2anc 584	. . . . 5 ⊢ (𝜑 → dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} = {(Base‘ndx), (Hom ‘ndx)})
16		estrres.x	. . . . . 6 ⊢ (𝜑 → · ∈ 𝑌)
17		dmsnopg 6215	. . . . . 6 ⊢ ( · ∈ 𝑌 → dom {⟨(comp‘ndx), · ⟩} = {(comp‘ndx)})
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → dom {⟨(comp‘ndx), · ⟩} = {(comp‘ndx)})
19	15, 18	uneq12d 4151	. . . 4 ⊢ (𝜑 → (dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩} ∪ dom {⟨(comp‘ndx), · ⟩}) = ({(Base‘ndx), (Hom ‘ndx)} ∪ {(comp‘ndx)}))
20	9, 11, 19	3eqtrd 2773	. . 3 ⊢ (𝜑 → dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} = ({(Base‘ndx), (Hom ‘ndx)} ∪ {(comp‘ndx)}))
21		estrres.c	. . . 4 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
22	21	dmeqd 5898	. . 3 ⊢ (𝜑 → dom 𝐶 = dom {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
23		df-tp 4613	. . . 4 ⊢ {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)} = ({(Base‘ndx), (Hom ‘ndx)} ∪ {(comp‘ndx)})
24	23	a1i 11	. . 3 ⊢ (𝜑 → {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)} = ({(Base‘ndx), (Hom ‘ndx)} ∪ {(comp‘ndx)}))
25	20, 22, 24	3eqtr4d 2779	. 2 ⊢ (𝜑 → dom 𝐶 = {(Base‘ndx), (Hom ‘ndx), (comp‘ndx)})
26	6, 25	eleqtrrd 2836	1 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∨ w3o 1085   = wceq 1539   ∈ wcel 2107  Vcvv 3464   ∪ cun 3931  {csn 4608  {cpr 4610  {ctp 4612  ⟨cop 4614  dom cdm 5667  ‘cfv 6542  ndxcnx 17213  Basecbs 17230  Hom chom 17288  compcco 17289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-tp 4613  df-op 4615  df-uni 4890  df-br 5126  df-dm 5677  df-iota 6495  df-fv 6550

This theorem is referenced by:  estrres  18159