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Theorem bnj518 34426
Description: Technical lemma for bnj852 34461. 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
bnj518.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj518.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj518.3 (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))
Assertion
Ref Expression
bnj518 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝑓,𝑖,𝑝,𝑦   𝑖,𝑛,𝑝   𝐴,𝑖,𝑝,𝑦   𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑛)   𝑅(𝑥,𝑓,𝑖,𝑛,𝑝)

Proof of Theorem bnj518
StepHypRef Expression
1 bnj518.3 . . . 4 (𝜏 ↔ (𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛))
2 bnj334 34253 . . . 4 ((𝜑𝜓𝑛 ∈ ω ∧ 𝑝𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
31, 2bitri 275 . . 3 (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛))
4 df-bnj17 34227 . . . 4 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) ↔ ((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛))
5 bnj518.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
6 bnj518.2 . . . . . 6 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
75, 6bnj517 34425 . . . . 5 ((𝑛 ∈ ω ∧ 𝜑𝜓) → ∀𝑝𝑛 (𝑓𝑝) ⊆ 𝐴)
87r19.21bi 3242 . . . 4 (((𝑛 ∈ ω ∧ 𝜑𝜓) ∧ 𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
94, 8sylbi 216 . . 3 ((𝑛 ∈ ω ∧ 𝜑𝜓𝑝𝑛) → (𝑓𝑝) ⊆ 𝐴)
103, 9sylbi 216 . 2 (𝜏 → (𝑓𝑝) ⊆ 𝐴)
11 ssel 3970 . . . 4 ((𝑓𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓𝑝) → 𝑦𝐴))
12 bnj93 34403 . . . . 5 ((𝑅 FrSe 𝐴𝑦𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
1312ex 412 . . . 4 (𝑅 FrSe 𝐴 → (𝑦𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
1411, 13sylan9r 508 . . 3 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
1514ralrimiv 3139 . 2 ((𝑅 FrSe 𝐴 ∧ (𝑓𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
1610, 15sylan2 592 1 ((𝑅 FrSe 𝐴𝜏) → ∀𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  Vcvv 3468  wss 3943  c0 4317   ciun 4990  suc csuc 6359  cfv 6536  ωcom 7851  w-bnj17 34226   predc-bnj14 34228   FrSe w-bnj15 34232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fv 6544  df-om 7852  df-bnj17 34227  df-bnj14 34229  df-bnj13 34231  df-bnj15 34233
This theorem is referenced by:  bnj535  34430  bnj546  34436
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