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Theorem bnj1018g 32475
 Description: Version of bnj1018 32476 with less disjoint variable conditions, but requiring ax-13 2379. (Contributed by Gino Giotto, 27-Mar-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1018.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1018.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1018.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1018.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1018.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1018.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1018.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1018.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1018.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1018.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1018.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1018.13 𝐷 = (ω ∖ {∅})
bnj1018.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1018.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1018.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1018.26 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
bnj1018.29 ((𝜃𝜒𝜏𝜂) → 𝜒″)
bnj1018.30 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Assertion
Ref Expression
bnj1018g ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝑓,𝑝   𝜑,𝑖   𝜃,𝑝
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑧,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1018g
StepHypRef Expression
1 df-bnj17 32197 . . 3 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 32218 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
3 bnj1018.29 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) → 𝜒″)
42, 3sylbir 238 . . . . . . 7 (((𝜃𝜒𝜂) ∧ 𝜏) → 𝜒″)
54ex 416 . . . . . 6 ((𝜃𝜒𝜂) → (𝜏𝜒″))
65eximdv 1918 . . . . 5 ((𝜃𝜒𝜂) → (∃𝑝𝜏 → ∃𝑝𝜒″))
7 bnj1018.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj1018.9 . . . . . 6 (𝜒′[𝑝 / 𝑛]𝜒)
9 bnj1018.12 . . . . . 6 (𝜒″[𝐺 / 𝑓]𝜒′)
10 bnj1018.14 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
11 bnj1018.16 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
127, 8, 9, 10, 11bnj985 32466 . . . . 5 (𝐺𝐵 ↔ ∃𝑝𝜒″)
136, 12syl6ibr 255 . . . 4 ((𝜃𝜒𝜂) → (∃𝑝𝜏𝐺𝐵))
1413imp 410 . . 3 (((𝜃𝜒𝜂) ∧ ∃𝑝𝜏) → 𝐺𝐵)
151, 14sylbi 220 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → 𝐺𝐵)
16 bnj1019 32291 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
17 bnj1018.30 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
1817simp3d 1141 . . . . 5 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
19 bnj1018.26 . . . . . . 7 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
2019bnj1235 32316 . . . . . 6 (𝜒″𝐺 Fn 𝑝)
21 fndm 6441 . . . . . 6 (𝐺 Fn 𝑝 → dom 𝐺 = 𝑝)
223, 20, 213syl 18 . . . . 5 ((𝜃𝜒𝜏𝜂) → dom 𝐺 = 𝑝)
2318, 22eleqtrrd 2855 . . . 4 ((𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2423exlimiv 1931 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2516, 24sylbir 238 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → suc 𝑖 ∈ dom 𝐺)
26 bnj1018.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27 bnj1018.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
28 bnj1018.13 . . 3 𝐷 = (ω ∖ {∅})
2911bnj918 32277 . . 3 𝐺 ∈ V
30 vex 3413 . . . 4 𝑖 ∈ V
3130sucex 7531 . . 3 suc 𝑖 ∈ V
3226, 27, 28, 10, 29, 31bnj1015 32474 . 2 ((𝐺𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
3315, 25, 32syl2anc 587 1 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  Vcvv 3409  [wsbc 3698   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  ∅c0 4227  {csn 4525  ⟨cop 4531  ∪ ciun 4886  dom cdm 5528  suc csuc 6176   Fn wfn 6335  ‘cfv 6340  ωcom 7585   ∧ w-bnj17 32196   predc-bnj14 32198   FrSe w-bnj15 32202   trClc-bnj18 32204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-dm 5538  df-suc 6180  df-iota 6299  df-fn 6343  df-fv 6348  df-bnj17 32197  df-bnj18 32205 This theorem is referenced by: (None)
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