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Theorem ctiunctlemuom 11960
Description: Lemma for ctiunct 11964. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som (𝜑𝑆 ⊆ ω)
ctiunct.sdc (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)
ctiunct.f (𝜑𝐹:𝑆onto𝐴)
ctiunct.tom ((𝜑𝑥𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc ((𝜑𝑥𝐴) → ∀𝑛 ∈ ω DECID 𝑛𝑇)
ctiunct.g ((𝜑𝑥𝐴) → 𝐺:𝑇onto𝐵)
ctiunct.j (𝜑𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
Assertion
Ref Expression
ctiunctlemuom (𝜑𝑈 ⊆ ω)

Proof of Theorem ctiunctlemuom
StepHypRef Expression
1 ctiunct.u . . 3 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
2 ssrab2 3182 . . 3 {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)} ⊆ ω
31, 2eqsstri 3129 . 2 𝑈 ⊆ ω
43a1i 9 1 (𝜑𝑈 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 819   = wceq 1331  wcel 1480  wral 2416  {crab 2420  csb 3003  wss 3071  ωcom 4504   × cxp 4537  ontowfo 5121  1-1-ontowf1o 5122  cfv 5123  1st c1st 6036  2nd c2nd 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-in 3077  df-ss 3084
This theorem is referenced by:  ctiunct  11964
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