Theorem ctiunctlemuom 12678

Description: Lemma for ctiunct 12682. (Contributed by Jim Kingdon, 28-Oct-2023.)

Hypotheses

Ref	Expression
ctiunct.som	⊢ (𝜑 → 𝑆 ⊆ ω)
ctiunct.sdc	⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
ctiunct.f	⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
ctiunct.tom	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
ctiunct.g	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)
ctiunct.j	⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u	⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}

Assertion

Ref	Expression
ctiunctlemuom	⊢ (𝜑 → 𝑈 ⊆ ω)

Proof of Theorem ctiunctlemuom

Step	Hyp	Ref	Expression
1		ctiunct.u	. . 3 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}
2		ssrab2 3269	. . 3 ⊢ {𝑧 ∈ ω ∣ ((1st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)} ⊆ ω
3	1, 2	eqsstri 3216	. 2 ⊢ 𝑈 ⊆ ω
4	3	a1i 9	1 ⊢ (𝜑 → 𝑈 ⊆ ω)

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479  ⦋csb 3084   ⊆ wss 3157  ωcom 4627   × cxp 4662  –onto→wfo 5257  –1-1-onto→wf1o 5258  ‘cfv 5259  1^st c1st 6205  2^nd c2nd 6206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-in 3163  df-ss 3170

This theorem is referenced by:  ctiunct  12682