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Theorem bnj929 30706
Description: Technical lemma for bnj69 30778. 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 30520 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑)
2 bnj334 30478 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑))
3 bnj257 30472 . . . . . . 7 ((𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
42, 3bitri 264 . . . . . 6 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛))
5 bnj345 30479 . . . . . 6 ((𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑))
6 bnj253 30469 . . . . . 6 ((𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
74, 5, 63bitri 286 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) ∧ 𝑛𝐷𝜑))
87simp1bi 1074 . . . 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 30539 . . . . 5 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
1211bnj930 30540 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → Fun 𝐺)
138, 12syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → Fun 𝐺)
149bnj931 30541 . . . 4 𝑓𝐺
1514a1i 11 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝑓𝐺)
16 bnj268 30474 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
17 bnj253 30469 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1816, 17bitr3i 266 . . . . 5 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ ((𝑛𝐷𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛𝜑))
1918simp1bi 1074 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑛𝐷𝑓 Fn 𝑛))
20 fndm 5950 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21 bnj929.10 . . . . . 6 𝐷 = (ω ∖ {∅})
2221bnj529 30511 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
23 eleq2 2693 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2423biimpar 502 . . . . 5 ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
2520, 22, 24syl2anr 495 . . . 4 ((𝑛𝐷𝑓 Fn 𝑛) → ∅ ∈ dom 𝑓)
2619, 25syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → ∅ ∈ dom 𝑓)
2713, 15, 26bnj1502 30618 . 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 30536 . . 3 𝐺 ∈ V
3228, 29, 30, 31bnj934 30705 . 2 ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
331, 27, 32syl2anc 692 1 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  [wsbc 3422  cdif 3557  cun 3558  wss 3560  c0 3896  {csn 4153  cop 4159  dom cdm 5079  suc csuc 5687  Fun wfun 5844   Fn wfn 5845  cfv 5850  ωcom 7013  w-bnj17 30451   predc-bnj14 30453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903  ax-reg 8442
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-om 7014  df-bnj17 30452
This theorem is referenced by:  bnj944  30708
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