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Theorem bnj938 34951
Description: Technical lemma for bnj69 35024. 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
bnj938.1 𝐷 = (ω ∖ {∅})
bnj938.2 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj938.3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj938.4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj938.5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj938 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝
Allowed substitution hints:   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑓,𝑚,𝑛)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑓,𝑚,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj938
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2823 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21bnj706 34768 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → ∃𝑥 𝑥 = 𝑋)
3 bnj291 34725 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑋𝐴))
43simplbi 497 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → (𝑅 FrSe 𝐴𝜏𝜎))
5 bnj602 34929 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
65eqeq2d 2748 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
7 bnj938.4 . . . . . . . . 9 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
86, 7bitr4di 289 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑′))
983anbi2d 1443 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚𝜑′𝜓′)))
10 bnj938.2 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
119, 10bitr4di 289 . . . . . 6 (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ 𝜏))
12113anbi2d 1443 . . . . 5 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) ↔ (𝑅 FrSe 𝐴𝜏𝜎)))
134, 12imbitrrid 246 . . . 4 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → (𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎)))
14 bnj938.1 . . . . 5 𝐷 = (ω ∖ {∅})
15 biid 261 . . . . 5 ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′))
16 bnj938.3 . . . . 5 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
17 biid 261 . . . . 5 ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
18 bnj938.5 . . . . 5 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑚 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1914, 15, 16, 17, 18bnj546 34910 . . . 4 ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
2013, 19syl6 35 . . 3 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
2120exlimiv 1930 . 2 (∃𝑥 𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V))
222, 21mpcom 38 1 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  cdif 3948  c0 4333  {csn 4626   ciun 4991  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887  w-bnj17 34700   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fv 6569  df-om 7888  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707
This theorem is referenced by:  bnj944  34952  bnj969  34960
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