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Theorem bnj938 31455
Description: Technical lemma for bnj69 31526. 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 3368 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21bnj706 31272 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → ∃𝑥 𝑥 = 𝑋)
3 bnj291 31228 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑋𝐴))
43simplbi 491 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → (𝑅 FrSe 𝐴𝜏𝜎))
5 bnj602 31433 . . . . . . . . . 10 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
65eqeq2d 2775 . . . . . . . . 9 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)))
7 bnj938.4 . . . . . . . . 9 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
86, 7syl6bbr 280 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ 𝜑′))
983anbi2d 1565 . . . . . . 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, 10syl6bbr 280 . . . . . 6 (𝑥 = 𝑋 → ((𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ↔ 𝜏))
12113anbi2d 1565 . . . . 5 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴 ∧ (𝑓 Fn 𝑚 ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ 𝜓′) ∧ 𝜎) ↔ (𝑅 FrSe 𝐴𝜏𝜎)))
134, 12syl5ibr 237 . . . 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 252 . . . . 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 252 . . . . 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 31414 . . . 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 2025 . 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 197  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  Vcvv 3350  cdif 3729  c0 4079  {csn 4334   ciun 4676  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31203   predc-bnj14 31205   FrSe w-bnj15 31209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-bnj17 31204  df-bnj14 31206  df-bnj13 31208  df-bnj15 31210
This theorem is referenced by:  bnj944  31456  bnj969  31464
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