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Theorem bnj1121 31385
 Description: Technical lemma for bnj69 31410. 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 2205 . . . . 5 (𝜒 → ∃𝑛𝜒)
21bnj707 31157 . . . 4 ((𝜃𝜏𝜒𝜁) → ∃𝑛𝜒)
3 bnj1121.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1121.7 . . . . 5 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
53, 4bnj1083 31378 . . . 4 (𝑓𝐾 ↔ ∃𝑛𝜒)
62, 5sylibr 224 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑓𝐾)
7 bnj1121.4 . . . . . 6 (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))
87simplbi 479 . . . . 5 (𝜁𝑖𝑛)
98bnj708 31158 . . . 4 ((𝜃𝜏𝜒𝜁) → 𝑖𝑛)
103bnj1235 31207 . . . . . 6 (𝜒𝑓 Fn 𝑛)
1110bnj707 31157 . . . . 5 ((𝜃𝜏𝜒𝜁) → 𝑓 Fn 𝑛)
12 fndm 6130 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
1311, 12syl 17 . . . 4 ((𝜃𝜏𝜒𝜁) → dom 𝑓 = 𝑛)
149, 13eleqtrrd 2852 . . 3 ((𝜃𝜏𝜒𝜁) → 𝑖 ∈ dom 𝑓)
15 bnj1121.6 . . . . 5 ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)
1615, 9bnj1294 31220 . . . 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 671 . 2 ((𝜃𝜏𝜒𝜁) → (𝑓𝑖) ⊆ 𝐵)
207simprbi 478 . . 3 (𝜁𝑧 ∈ (𝑓𝑖))
2120bnj708 31158 . 2 ((𝜃𝜏𝜒𝜁) → 𝑧 ∈ (𝑓𝑖))
2219, 21sseldd 3751 1 ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630  ∃wex 1851   ∈ wcel 2144  {cab 2756  ∀wral 3060  ∃wrex 3061  Vcvv 3349   ⊆ wss 3721  dom cdm 5249   Fn wfn 6026  ‘cfv 6031   ∧ w-bnj17 31086   predc-bnj14 31088   FrSe w-bnj15 31092   TrFow-bnj19 31096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-ral 3065  df-rex 3066  df-in 3728  df-ss 3735  df-fn 6034  df-bnj17 31087 This theorem is referenced by:  bnj1030  31387
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