Theorem bnj929 35233

Description: Technical lemma for bnj69 35307. 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	⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 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

Step	Hyp	Ref	Expression
1		bnj645 35048	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑)
2		bnj334 35011	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
3		bnj257 35005	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
4	2, 3	bitri 277	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
5		bnj345 35012	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
6		bnj253 35002	. . . . . 6 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
7	4, 5, 6	3bitri 299	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
8	7	simp1bi 1159	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
9		bnj929.13	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
10		bnj929.50	. . . . . 6 ⊢ 𝐶 ∈ V
11	9, 10	bnj927 35067	. . . . 5 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
12	11	fnfund 6624	. . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → Fun 𝐺)
13	8, 12	syl 17	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → Fun 𝐺)
14	9	bnj931 35068	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	14	a1i 11	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝑓 ⊆ 𝐺)
16		bnj268 35007	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
17		bnj253 35002	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
18	16, 17	bitr3i 279	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
19	18	simp1bi 1159	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛))
20		fndm 6626	. . . . 5 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21		bnj929.10	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
22	21	bnj529 35039	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
23		eleq2 2853	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
24	23	biimpar 481	. . . . 5 ⊢ ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
25	20, 22, 24	syl2anr 606	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) → ∅ ∈ dom 𝑓)
26	19, 25	syl 17	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → ∅ ∈ dom 𝑓)
27	13, 15, 26	bnj1502 35145	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (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′)
31	9	bnj918 35064	. . 3 ⊢ 𝐺 ∈ V
32	28, 29, 30, 31	bnj934 35232	. 2 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
33	1, 27, 32	syl2anc 593	1 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  Vcvv 3456  [wsbc 3746   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4584  ⟨cop 4590  dom cdm 5649  suc csuc 6350  Fun wfun 6517   Fn wfn 6518  ‘cfv 6523  ωcom 7848   ∧ w-bnj17 34984   predc-bnj14 34986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-res 5661  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-om 7849  df-bnj17 34985

This theorem is referenced by:  bnj944  35235