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 32107
Description: Technical lemma for bnj69 32179. 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 31920 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑)
2 bnj334 31882 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑))
3 bnj257 31876 . . . . . . 7 ((𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
42, 3bitri 276 . . . . . 6 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
5 bnj345 31883 . . . . . 6 ((𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑))
6 bnj253 31873 . . . . . 6 ((𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
74, 5, 63bitri 298 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
87simp1bi 1137 . . . 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 31939 . . . . 5 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
1211bnj930 31940 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → Fun 𝐺)
138, 12syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → Fun 𝐺)
149bnj931 31941 . . . 4 𝑓𝐺
1514a1i 11 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝑓𝐺)
16 bnj268 31878 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
17 bnj253 31873 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1816, 17bitr3i 278 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1918simp1bi 1137 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑛𝐷𝑓 Fn 𝑛))
20 fndm 6448 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21 bnj929.10 . . . . . 6 𝐷 = (ω ∖ {∅})
2221bnj529 31911 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
23 eleq2 2898 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2423biimpar 478 . . . . 5 ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
2520, 22, 24syl2anr 596 . . . 4 ((𝑛𝐷𝑓 Fn 𝑛) → ∅ ∈ dom 𝑓)
2619, 25syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → ∅ ∈ dom 𝑓)
2713, 15, 26bnj1502 32019 . 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 31936 . . 3 𝐺 ∈ V
3228, 29, 30, 31bnj934 32106 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  [wsbc 3769  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557  cop 4563  dom cdm 5548  suc csuc 6186  Fun wfun 6342   Fn wfn 6343  cfv 6348  ωcom 7569  w-bnj17 31855   predc-bnj14 31857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-om 7570  df-bnj17 31856
This theorem is referenced by:  bnj944  32109
  Copyright terms: Public domain W3C validator