Theorem bnj944 30262

Description: Technical lemma for bnj69 30332. 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 472	. . . 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 30076	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	2, 3	sylbi 206	. . . . . . 7 ⊢ (𝜒 → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		bnj944.14	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	4, 5	sylibr 223	. . . . . 6 ⊢ (𝜒 → 𝜏)
7	6	3ad2ant1 1075	. . . . 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 30128	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
10		vex 3176	. . . . . . . 8 ⊢ 𝑚 ∈ V
11	10	bnj216 30054	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → 𝑚 ∈ 𝑛)
12		id 22	. . . . . . 7 ⊢ (𝑝 = suc 𝑛 → 𝑝 = suc 𝑛)
13	9, 11, 12	3anim123i 1240	. . . . . 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 1040	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 263	. . . . . 6 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑚 ∈ 𝑛 ∧ 𝑝 = suc 𝑛))
17	13, 16	sylibr 223	. . . . 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 30023	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜏 ∧ 𝜎))
20	1, 8, 18, 19	syl3anbrc 1239	. . 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 30261	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
26	21, 25	syl5eqel 2692	. . 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 30075	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
29	2, 28	sylbi 206	. . . . 5 ⊢ (𝜒 → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
30	29	3ad2ant1 1075	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑))
31		simp3 1056	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
32		bnj291 30030	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝑝 = suc 𝑛))
33	30, 31, 32	sylanbrc 695	. . 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 4341	. . . . . . . . 9 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
38	37	sneqd 4137	. . . . . . . 8 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
39	38	uneq2d 3729	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
40	36, 39	syl5eq 2656	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
41	40	sbceq1d 3407	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ([𝐺 / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
42	35, 41	syl5bb 271	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝜑″ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′))
43	42	imbi2d 329	. . 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 250	. . . 4 ⊢ ([(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′ ↔ [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
46		eqid 2610	. . . 4 ⊢ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
47		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
48	47	elimel 4100	. . . 4 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V
49	23, 44, 45, 22, 46, 48	bnj929 30260	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → [(𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) / 𝑓]𝜑′)
50	43, 49	dedth 4089	. 2 ⊢ (𝐶 ∈ V → ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) → 𝜑″))
51	27, 34, 50	sylc 63	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012

This theorem is referenced by:  bnj910  30272