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Theorem bnj1121 30788
Description: Technical lemma for bnj69 30813. 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
bnj1121.1 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))
bnj1121.2 (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1121.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1121.4 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
bnj1121.5 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1121.6 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
bnj1121.7 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1121 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)

Proof of Theorem bnj1121
StepHypRef Expression
1 19.8a 2049 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 30560 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 30781 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 224 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 476 . . . . 5 (𝜁𝑖𝑛)
98bnj708 30561 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 30610 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 30560 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
12 fndm 5953 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
1311, 12syl 17 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
149, 13eleqtrrd 2701 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
15 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1615, 9bnj1294 30623 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝜂)
17 bnj1121.5 . . . 4 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
1816, 17sylib 208 . . 3 ((𝜃𝜏𝜒𝜁) → ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
196, 14, 18mp2and 714 . 2 ((𝜃𝜏𝜒𝜁) → (𝑓𝑖) ⊆ 𝐵)
207simprbi 480 . . 3 (𝜁𝑧 ∈ (𝑓𝑖))
2120bnj708 30561 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2219, 21sseldd 3588 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  wss 3559  dom cdm 5079   Fn wfn 5847  cfv 5852  w-bnj17 30486   predc-bnj14 30488   FrSe w-bnj15 30492   TrFow-bnj19 30496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-rex 2913  df-in 3566  df-ss 3573  df-fn 5855  df-bnj17 30487
This theorem is referenced by:  bnj1030  30790
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