Theorem bnj929 31006

Description: Technical lemma for bnj69 31078. 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 30820	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑)
2		bnj334 30779	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
3		bnj257 30773	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
4	2, 3	bitri 264	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ (𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛))
5		bnj345 30780	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑 ∧ 𝑝 = suc 𝑛) ↔ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
6		bnj253 30770	. . . . . 6 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
7	4, 5, 6	3bitri 286	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ∧ 𝑛 ∈ 𝐷 ∧ 𝜑))
8	7	simp1bi 1076	. . . 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 30839	. . . . 5 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
12	11	bnj930 30840	. . . 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 30841	. . . 4 ⊢ 𝑓 ⊆ 𝐺
15	14	a1i 11	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝑓 ⊆ 𝐺)
16		bnj268 30775	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
17		bnj253 30770	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
18	16, 17	bitr3i 266	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛) ∧ 𝑝 = suc 𝑛 ∧ 𝜑))
19	18	simp1bi 1076	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛))
20		fndm 5990	. . . . 5 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21		bnj929.10	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
22	21	bnj529 30811	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
23		eleq2 2690	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
24	23	biimpar 502	. . . . 5 ⊢ ((dom 𝑓 = 𝑛 ∧ ∅ ∈ 𝑛) → ∅ ∈ dom 𝑓)
25	20, 22, 24	syl2anr 495	. . . 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 30918	. 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 30836	. . 3 ⊢ 𝐺 ∈ V
32	28, 29, 30, 31	bnj934 31005	. 2 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
33	1, 27, 32	syl2anc 693	1 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  [wsbc 3435   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  dom cdm 5114  suc csuc 5725  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  ωcom 7065   ∧ w-bnj17 30752   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj944  31008