Theorem bnj929 32816

Description: Technical lemma for bnj69 32890. 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 32630	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑)
2		bnj334 32592	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
3		bnj257 32586	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
4	2, 3	bitri 274	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
5		bnj345 32593	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
6		bnj253 32583	. . . . . 6 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
7	4, 5, 6	3bitri 296	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
8	7	simp1bi 1143	. . . 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 32649	. . . . 5 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
12	11	fnfund 6518	. . . 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 32650	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	14	a1i 11	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝑓 ⊆ 𝐺)
16		bnj268 32588	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
17		bnj253 32583	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
18	16, 17	bitr3i 276	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
19	18	simp1bi 1143	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛))
20		fndm 6520	. . . . 5 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21		bnj929.10	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
22	21	bnj529 32621	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
23		eleq2 2827	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
24	23	biimpar 477	. . . . 5 ⊢ ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
25	20, 22, 24	syl2anr 596	. . . 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 32728	. 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 32646	. . 3 ⊢ 𝐺 ∈ V
32	28, 29, 30, 31	bnj934 32815	. 2 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
33	1, 27, 32	syl2anc 583	1 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  [wsbc 3711   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  dom cdm 5580  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  ωcom 7687   ∧ w-bnj17 32565   predc-bnj14 32567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-om 7688  df-bnj17 32566

This theorem is referenced by:  bnj944  32818