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Theorem bnj929 31816
Description: Technical lemma for bnj69 31888. 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 31630 . 2 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝜑)
2 bnj334 31592 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) ↔ (𝑓 Fn 𝑛𝑛𝐷𝑝 = suc 𝑛𝜑))
3 bnj257 31586 . . . . . . 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 31593 . . . . . 6 ((𝑓 Fn 𝑛𝑛𝐷𝜑𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛𝑓 Fn 𝑛𝑛𝐷𝜑))
6 bnj253 31583 . . . . . 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 1138 . . . 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 31649 . . . . 5 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
1211bnj930 31650 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → Fun 𝐺)
138, 12syl 17 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → Fun 𝐺)
149bnj931 31651 . . . 4 𝑓𝐺
1514a1i 11 . . 3 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → 𝑓𝐺)
16 bnj268 31588 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝑝 = suc 𝑛𝜑) ↔ (𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑))
17 bnj253 31583 . . . . . 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 1138 . . . 4 ((𝑛𝐷𝑝 = suc 𝑛𝑓 Fn 𝑛𝜑) → (𝑛𝐷𝑓 Fn 𝑛))
20 fndm 6328 . . . . 5 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21 bnj929.10 . . . . . 6 𝐷 = (ω ∖ {∅})
2221bnj529 31621 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
23 eleq2 2870 . . . . . 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 31728 . 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 31646 . . 3 𝐺 ∈ V
3228, 29, 30, 31bnj934 31815 . 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 1080   = wceq 1522  wcel 2080  Vcvv 3436  [wsbc 3707  cdif 3858  cun 3859  wss 3861  c0 4213  {csn 4474  cop 4480  dom cdm 5446  suc csuc 6071  Fun wfun 6222   Fn wfn 6223  cfv 6228  ωcom 7439  w-bnj17 31565   predc-bnj14 31567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224  ax-un 7322  ax-reg 8905
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-res 5458  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-fv 6236  df-om 7440  df-bnj17 31566
This theorem is referenced by:  bnj944  31818
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