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Theorem ctiunctlemu2nd 12383
Description: Lemma for ctiunct 12388. (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
ctiunctlemu2nd (𝜑 → (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)
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 ctiunctlemu2nd
StepHypRef Expression
1 ctiunctlem.n . . . 4 (𝜑𝑁𝑈)
2 2fveq3 5499 . . . . . . 7 (𝑧 = 𝑁 → (1st ‘(𝐽𝑧)) = (1st ‘(𝐽𝑁)))
32eleq1d 2239 . . . . . 6 (𝑧 = 𝑁 → ((1st ‘(𝐽𝑧)) ∈ 𝑆 ↔ (1st ‘(𝐽𝑁)) ∈ 𝑆))
4 2fveq3 5499 . . . . . . 7 (𝑧 = 𝑁 → (2nd ‘(𝐽𝑧)) = (2nd ‘(𝐽𝑁)))
52fveq2d 5498 . . . . . . . 8 (𝑧 = 𝑁 → (𝐹‘(1st ‘(𝐽𝑧))) = (𝐹‘(1st ‘(𝐽𝑁))))
65csbeq1d 3056 . . . . . . 7 (𝑧 = 𝑁(𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 = (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)
74, 6eleq12d 2241 . . . . . 6 (𝑧 = 𝑁 → ((2nd ‘(𝐽𝑧)) ∈ (𝐹‘(1st ‘(𝐽𝑧))) / 𝑥𝑇 ↔ (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇))
83, 7anbi12d 470 . . . . 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 2889 . . . 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))
1312simprd 113 1 (𝜑 → (2nd ‘(𝐽𝑁)) ∈ (𝐹‘(1st ‘(𝐽𝑁))) / 𝑥𝑇)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  {crab 2452  csb 3049  wss 3121  ωcom 4572   × cxp 4607  ontowfo 5194  1-1-ontowf1o 5195  cfv 5196  1st c1st 6115  2nd c2nd 6116
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204
This theorem is referenced by:  ctiunctlemf  12386
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