Theorem bnj944 32818

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
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 482	. . . 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 32632	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	2, 3	sylbi 216	. . . . . . 7 ⊢ (𝜒 → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		bnj944.14	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	4, 5	sylibr 233	. . . . . 6 ⊢ (𝜒 → 𝜏)
7	6	3ad2ant1 1131	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜏)
8	7	adantl 481	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜏)
9	2	bnj1232 32683	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
10		vex 3426	. . . . . . . 8 ⊢ 𝑚 ∈ V
11	10	bnj216 32611	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
12		id 22	. . . . . . 7 ⊢ (𝑝 = suc 𝑛 → 𝑝 = suc 𝑛)
13	9, 11, 12	3anim123i 1149	. . . . . 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 1097	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 274	. . . . . 6 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
17	13, 16	sylibr 233	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜎)
18	17	adantl 481	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜎)
19		bnj253 32583	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜏 ∧ 𝜎))
20	1, 8, 18, 19	syl3anbrc 1341	. . 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 32817	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
26	21, 25	eqeltrid 2843	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐶 ∈ V)
27	20, 26	syl 17	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
28		bnj658 32631	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
29	2, 28	sylbi 216	. . . . 5 ⊢ (𝜒 → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
30	29	3ad2ant1 1131	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
31		simp3 1136	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
32		bnj291 32590	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝑝 = suc 𝑛))
33	30, 31, 32	sylanbrc 582	. . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
34	33	adantl 481	. 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 4802	. . . . . . . . 9 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
38	37	sneqd 4570	. . . . . . . 8 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
39	38	uneq2d 4093	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
40	36, 39	syl5eq 2791	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
41	40	sbceq1d 3716	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
42	35, 41	syl5bb 282	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
43	42	imbi2d 340	. . 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 260	. . . 4 ⊢ ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46		eqid 2738	. . . 4 ⊢ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
48	47	elimel 4525	. . . 4 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V
49	23, 44, 45, 22, 46, 48	bnj929 32816	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
50	43, 49	dedth 4514	. 2 ⊢ (𝐶 ∈ V → ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″))
51	27, 34, 50	sylc 65	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  [wsbc 3711   ∖ cdif 3880   ∪ cun 3881  ∅c0 4253  ifcif 4456  {csn 4558  ⟨cop 4564  ∪ ciun 4921  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687   ∧ w-bnj17 32565   predc-bnj14 32567   FrSe w-bnj15 32571

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-rep 5205  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-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  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-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  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-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-bnj17 32566  df-bnj14 32568  df-bnj13 32570  df-bnj15 32572

This theorem is referenced by:  bnj910  32828