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Theorem ctiunctlemu1st 11936
 Description: Lemma for ctiunct 11942. (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)}
ctiunctlem.n (𝜑𝑁𝑈)
Assertion
Ref Expression
ctiunctlemu1st (𝜑 → (1st ‘(𝐽𝑁)) ∈ 𝑆)
Distinct variable groups:   𝑧,𝐹   𝑧,𝐽   𝑧,𝑁   𝑧,𝑆   𝑧,𝑇   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑛)   𝐴(𝑥,𝑧,𝑛)   𝐵(𝑥,𝑧,𝑛)   𝑆(𝑥,𝑛)   𝑇(𝑥,𝑛)   𝑈(𝑥,𝑧,𝑛)   𝐹(𝑥,𝑛)   𝐺(𝑥,𝑧,𝑛)   𝐽(𝑥,𝑛)   𝑁(𝑥,𝑛)

Proof of Theorem ctiunctlemu1st
StepHypRef Expression
1 ctiunctlem.n . . . 4 (𝜑𝑁𝑈)
2 2fveq3 5419 . . . . . . 7 (𝑧 = 𝑁 → (1st ‘(𝐽𝑧)) = (1st ‘(𝐽𝑁)))
32eleq1d 2206 . . . . . 6 (𝑧 = 𝑁 → ((1st ‘(𝐽𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽𝑁)) ∈ 𝑆))
4 2fveq3 5419 . . . . . . 7 (𝑧 = 𝑁 → (2nd ‘(𝐽𝑧)) = (2nd ‘(𝐽𝑁)))
52fveq2d 5418 . . . . . . . 8 (𝑧 = 𝑁 → (𝐹‘(1st ‘(𝐽𝑧))) = (𝐹‘(1st ‘(𝐽𝑁))))
65csbeq1d 3005 . . . . . . 7 (𝑧 = 𝑁(𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 = (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)
74, 6eleq12d 2208 . . . . . 6 (𝑧 = 𝑁 → ((2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 ↔ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇))
83, 7anbi12d 464 . . . . 5 (𝑧 = 𝑁 → (((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇) ↔ ((1st ‘(𝐽𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)))
9 ctiunct.u . . . . 5 𝑈 = {𝑧 ∈ ω ∣ ((1st ‘(𝐽𝑧)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇)}
108, 9elrab2 2838 . . . 4 (𝑁𝑈 ↔ (𝑁 ∈ ω ∧ ((1st ‘(𝐽𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)))
111, 10sylib 121 . . 3 (𝜑 → (𝑁 ∈ ω ∧ ((1st ‘(𝐽𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)))
1211simprd 113 . 2 (𝜑 → ((1st ‘(𝐽𝑁)) ∈ 𝑆 ∧ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇))
1312simpld 111 1 (𝜑 → (1st ‘(𝐽𝑁)) ∈ 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2414  {crab 2418  ⦋csb 2998   ⊆ wss 3066  ωcom 4499   × cxp 4532  –onto→wfo 5116  –1-1-onto→wf1o 5117  ‘cfv 5118  1st c1st 6029  2nd c2nd 6030 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 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126 This theorem is referenced by:  ctiunctlemf  11940
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