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Theorem bnj1030 31362
Description: Technical lemma for bnj69 31385. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1030.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1030.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1030.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1030.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1030.5 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1030.6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1030.7 𝐷 = (ω ∖ {∅})
bnj1030.8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1030.9 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1030.10 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
bnj1030.11 (𝜑′[𝑗 / 𝑖]𝜑)
bnj1030.12 (𝜓′[𝑗 / 𝑖]𝜓)
bnj1030.13 (𝜒′[𝑗 / 𝑖]𝜒)
bnj1030.14 (𝜃′[𝑗 / 𝑖]𝜃)
bnj1030.15 (𝜏′[𝑗 / 𝑖]𝜏)
bnj1030.16 (𝜁′[𝑗 / 𝑖]𝜁)
bnj1030.17 (𝜂′[𝑗 / 𝑖]𝜂)
bnj1030.18 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
bnj1030.19 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
Assertion
Ref Expression
bnj1030 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑛,𝑦   𝑧,𝐴,𝑓,𝑖,𝑛   𝐵,𝑓,𝑖,𝑛,𝑦   𝑧,𝐵   𝐷,𝑖,𝑗   𝑅,𝑓,𝑖,𝑗,𝑛,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑛,𝑦   𝑧,𝑋   𝜒,𝑗   𝜂,𝑗   𝜏,𝑓,𝑖,𝑗,𝑛   𝜃,𝑓,𝑖,𝑗,𝑛   𝜑,𝑖   𝜏,𝑧   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦)   𝜏(𝑦)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜁(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜎(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜌(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑗)   𝐷(𝑦,𝑧,𝑓,𝑛)   𝐾(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝑋(𝑗)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜃′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜏′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜁′(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)   𝜑0(𝑦,𝑧,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1030
StepHypRef Expression
1 bnj1030.1 . 2 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1030.2 . 2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1030.3 . 2 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1030.4 . 2 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
5 bnj1030.5 . 2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6 bnj1030.6 . 2 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
7 bnj1030.7 . 2 𝐷 = (ω ∖ {∅})
8 bnj1030.8 . 2 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
9 19.23vv 2021 . . . . 5 (∀𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
109albii 1896 . . . 4 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ ∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
11 19.23v 2020 . . . 4 (∀𝑓(∃𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
1210, 11bitri 264 . . 3 (∀𝑓𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵) ↔ (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵))
13 bnj1030.9 . . . . 5 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
147bnj1071 31352 . . . . . . . 8 (𝑛𝐷 → E Fr 𝑛)
153, 14bnj769 31139 . . . . . . 7 (𝜒 → E Fr 𝑛)
1615bnj707 31132 . . . . . 6 ((𝜃𝜏𝜒𝜁) → E Fr 𝑛)
17 bnj1030.10 . . . . . . 7 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
18 bnj1030.17 . . . . . . 7 (𝜂′[𝑗 / 𝑖]𝜂)
19 bnj1030.18 . . . . . . 7 (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))
20 bnj1030.19 . . . . . . 7 (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))
212, 8, 13, 18bnj1123 31361 . . . . . . . . . 10 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
222, 3, 5, 7, 19, 20, 21bnj1118 31359 . . . . . . . . 9 𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
231, 3, 5bnj1097 31356 . . . . . . . . 9 ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)
2422, 23bnj1109 31164 . . . . . . . 8 𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)
2524, 2, 3bnj1093 31355 . . . . . . 7 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))
2613, 17, 18, 19, 20, 25bnj1090 31354 . . . . . 6 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))
27 vex 3343 . . . . . . 7 𝑛 ∈ V
2827, 17bnj110 31235 . . . . . 6 (( E Fr 𝑛 ∧ ∀𝑖𝑛 (𝜌𝜂)) → ∀𝑖𝑛 𝜂)
2916, 26, 28syl2anc 696 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
304, 5, 3, 6, 13, 29, 8bnj1121 31360 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3130gen2 1872 . . 3 𝑛𝑖((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
3212, 31mpgbi 1874 . 2 (∃𝑓𝑛𝑖(𝜃𝜏𝜒𝜁) → 𝑧𝐵)
331, 2, 3, 4, 5, 6, 7, 8, 32bnj1034 31345 1 ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  [wsbc 3576  cdif 3712  wss 3715  c0 4058  {csn 4321   ciun 4672   class class class wbr 4804   E cep 5178   Fr wfr 5222  dom cdm 5266  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7230  w-bnj17 31061   predc-bnj14 31063   FrSe w-bnj15 31067   trClc-bnj18 31069   TrFow-bnj19 31071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fn 6052  df-fv 6057  df-om 7231  df-bnj17 31062  df-bnj18 31070  df-bnj19 31072
This theorem is referenced by:  bnj1124  31363
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