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Theorem ctiunctlemuom 12437
Description: Lemma for ctiunct 12441. (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 3241 . . 3 {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)} ⊆ ω
31, 2eqsstri 3188 . 2 𝑈 ⊆ ω
43a1i 9 1 (𝜑𝑈 ⊆ ω)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  {crab 2459  csb 3058  wss 3130  ωcom 4590   × cxp 4625  ontowfo 5215  1-1-ontowf1o 5216  cfv 5217  1st c1st 6139  2nd c2nd 6140
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3136  df-ss 3143
This theorem is referenced by:  ctiunct  12441
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