Theorem bnj944 35335

Description: Technical lemma for bnj69 35407. 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
bnj944.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj944.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj944.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj944.4	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj944.7	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj944.10	⊢ 𝐷 = (ω ∖ {∅})
bnj944.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj944.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj944.14	⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj944.15	⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))

Assertion

Ref	Expression
bnj944	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛   𝑦,𝐴,𝑓,𝑖,𝑚   𝑅,𝑓,𝑖,𝑚,𝑛   𝑦,𝑅   𝑓,𝑋,𝑛

Allowed substitution hints:   𝜑(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜏(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜎(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐴(𝑝)   𝐶(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐷(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝑅(𝑝)   𝐺(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝑋(𝑦, 𝑖, 𝑚, 𝑝)   𝜑′(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑″(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj944

Step	Hyp	Ref	Expression
1		simpl 487	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
2		bnj944.3	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3		bnj667 35150	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	2, 3	sylbi 220	. . . . . . 7 ⊢ (𝜒 → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		bnj944.14	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	4, 5	sylibr 237	. . . . . 6 ⊢ (𝜒 → 𝜏)
7	6	3ad2ant1 1150	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜏)
8	7	adantl 486	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜏)
9	2	bnj1232 35200	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
10		vex 3458	. . . . . . . 8 ⊢ 𝑚 ∈ V
11	10	bnj216 35130	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
12		id 23	. . . . . . 7 ⊢ (𝑝 = suc 𝑛 → 𝑝 = suc 𝑛)
13	9, 11, 12	3anim123i 1168	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
14		bnj944.15	. . . . . . 7 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
15		3ancomb 1115	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 278	. . . . . 6 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
17	13, 16	sylibr 237	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜎)
18	17	adantl 486	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜎)
19		bnj253 35102	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜏 ∧ 𝜎))
20	1, 8, 18, 19	syl3anbrc 1361	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎))
21		bnj944.12	. . . 4 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
22		bnj944.10	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
23		bnj944.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24		bnj944.2	. . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
25	22, 5, 14, 23, 24	bnj938 35334	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
26	21, 25	eqeltrid 2866	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐶 ∈ V)
27	20, 26	syl 18	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
28		bnj658 35149	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
29	2, 28	sylbi 220	. . . . 5 ⊢ (𝜒 → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
30	29	3ad2ant1 1150	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
31		simp3 1155	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
32		bnj291 35109	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝑝 = suc 𝑛))
33	30, 31, 32	sylanbrc 594	. . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
34	33	adantl 486	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
35		bnj944.7	. . . . 5 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
36		bnj944.13	. . . . . . 7 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
37		opeq2 4838	. . . . . . . . 9 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
38	37	sneqd 4600	. . . . . . . 8 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
39	38	uneq2d 4121	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
40	36, 39	eqtrid 2809	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
41	40	sbceq1d 3748	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
42	35, 41	bitrid 286	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
43	42	imbi2d 343	. . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)))
44		bnj944.4	. . . 4 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
45		biid 264	. . . 4 ⊢ ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46		eqid 2762	. . . 4 ⊢ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47		0ex 5269	. . . . 5 ⊢ ∅ ∈ V
48	47	elimel 4556	. . . 4 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V
49	23, 44, 45, 22, 46, 48	bnj929 35333	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
50	43, 49	dedth 4545	. 2 ⊢ (𝐶 ∈ V → ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″))
51	27, 34, 50	sylc 66	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454  [wsbc 3743   ∖ cdif 3901   ∪ cun 3902  ∅c0 4285  ifcif 4486  {csn 4588  ⟨cop 4594  ∪ ciun 4955  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860   ∧ w-bnj17 35084   predc-bnj14 35086   FrSe w-bnj15 35090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7861  df-bnj17 35085  df-bnj14 35087  df-bnj13 35089  df-bnj15 35091

This theorem is used by:  bnj910  35345