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Theorem bnj929 34950
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
bnj929.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj929.4 (𝜑′[𝑝 / 𝑛]𝜑)
bnj929.7 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj929.10 𝐷 = (ω ∖ {∅})
bnj929.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj929.50 𝐶 ∈ V
Assertion
Ref Expression
bnj929 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″)
Distinct variable groups:   𝐴,𝑓,𝑛   𝑅,𝑓,𝑛   𝑓,𝑋,𝑛
Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝐴(𝑝)   𝐶(𝑓,𝑛,𝑝)   𝐷(𝑓,𝑛,𝑝)   𝑅(𝑝)   𝐺(𝑓,𝑛,𝑝)   𝑋(𝑝)   𝜑′(𝑓,𝑛,𝑝)   𝜑″(𝑓,𝑛,𝑝)

Proof of Theorem bnj929
StepHypRef Expression
1 bnj645 34764 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑)
2 bnj334 34727 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑))
3 bnj257 34721 . . . . . . 7 ((𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
42, 3bitri 275 . . . . . 6 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
5 bnj345 34728 . . . . . 6 ((𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑))
6 bnj253 34718 . . . . . 6 ((𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
74, 5, 63bitri 297 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
87simp1bi 1146 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑝 = suc 𝑛𝑓 Fn 𝑛))
9 bnj929.13 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10 bnj929.50 . . . . . 6 𝐶 ∈ V
119, 10bnj927 34783 . . . . 5 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
1211fnfund 6669 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → Fun 𝐺)
138, 12syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → Fun 𝐺)
149bnj931 34784 . . . 4 𝑓𝐺
1514a1i 11 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝑓𝐺)
16 bnj268 34723 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
17 bnj253 34718 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1816, 17bitr3i 277 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1918simp1bi 1146 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑛𝐷𝑓 Fn 𝑛))
20 fndm 6671 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21 bnj929.10 . . . . . 6 𝐷 = (ω ∖ {∅})
2221bnj529 34755 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
23 eleq2 2830 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2423biimpar 477 . . . . 5 ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
2520, 22, 24syl2anr 597 . . . 4 ((𝑛𝐷𝑓 Fn 𝑛) → ∅ ∈ dom 𝑓)
2619, 25syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → ∅ ∈ dom 𝑓)
2713, 15, 26bnj1502 34862 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝐺‘∅) = (𝑓‘∅))
28 bnj929.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
29 bnj929.4 . . 3 (𝜑′[𝑝 / 𝑛]𝜑)
30 bnj929.7 . . 3 (𝜑″[𝐺 / 𝑓]𝜑′)
319bnj918 34780 . . 3 𝐺 ∈ V
3228, 29, 30, 31bnj934 34949 . 2 ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
331, 27, 32syl2anc 584 1 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  cdif 3948  cun 3949  wss 3951  c0 4333  {csn 4626  cop 4632  dom cdm 5685  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  cfv 6561  ωcom 7887  w-bnj17 34700   predc-bnj14 34702
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-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
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-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-om 7888  df-bnj17 34701
This theorem is referenced by:  bnj944  34952
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