Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj929 Structured version   Visualization version   GIF version

Theorem bnj929 34929
Description: Technical lemma for bnj69 35003. 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 34743 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑)
2 bnj334 34706 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑))
3 bnj257 34700 . . . . . . 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 34707 . . . . . 6 ((𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑))
6 bnj253 34697 . . . . . 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 1144 . . . 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 34762 . . . . 5 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
1211fnfund 6670 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → Fun 𝐺)
138, 12syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → Fun 𝐺)
149bnj931 34763 . . . 4 𝑓𝐺
1514a1i 11 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝑓𝐺)
16 bnj268 34702 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
17 bnj253 34697 . . . . . 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 1144 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑛𝐷𝑓 Fn 𝑛))
20 fndm 6672 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21 bnj929.10 . . . . . 6 𝐷 = (ω ∖ {∅})
2221bnj529 34734 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
23 eleq2 2828 . . . . . 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 34841 . 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 34759 . . 3 𝐺 ∈ V
3228, 29, 30, 31bnj934 34928 . 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 1086   = wceq 1537  wcel 2106  Vcvv 3478  [wsbc 3791  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631  cop 4637  dom cdm 5689  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  cfv 6563  ωcom 7887  w-bnj17 34679   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-om 7888  df-bnj17 34680
This theorem is referenced by:  bnj944  34931
  Copyright terms: Public domain W3C validator