Theorem bnj929 35118

Description: Technical lemma for bnj69 35192. 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 34933	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑)
2		bnj334 34896	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
3		bnj257 34890	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
4	2, 3	bitri 275	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
5		bnj345 34897	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
6		bnj253 34887	. . . . . 6 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
7	4, 5, 6	3bitri 297	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
8	7	simp1bi 1146	. . . 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 34952	. . . . 5 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
12	11	fnfund 6603	. . . 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 34953	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	14	a1i 11	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝑓 ⊆ 𝐺)
16		bnj268 34892	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
17		bnj253 34887	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
18	16, 17	bitr3i 277	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
19	18	simp1bi 1146	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛))
20		fndm 6605	. . . . 5 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21		bnj929.10	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
22	21	bnj529 34924	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
23		eleq2 2826	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
24	23	biimpar 477	. . . . 5 ⊢ ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
25	20, 22, 24	syl2anr 598	. . . 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 35030	. 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 34949	. . 3 ⊢ 𝐺 ∈ V
32	28, 29, 30, 31	bnj934 35117	. 2 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
33	1, 27, 32	syl2anc 585	1 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  [wsbc 3742   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  dom cdm 5634  suc csuc 6329  Fun wfun 6496   Fn wfn 6497  ‘cfv 6502  ωcom 7820   ∧ w-bnj17 34869   predc-bnj14 34871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381  ax-un 7692  ax-reg 9511

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-ord 6330  df-on 6331  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-fv 6510  df-om 7821  df-bnj17 34870

This theorem is referenced by:  bnj944  35120