Theorem bnj910 35498

Description: Technical lemma for bnj69 35560. 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
bnj910.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj910.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj910.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj910.4	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj910.5	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj910.6	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj910.7	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj910.8	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj910.9	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj910.10	⊢ 𝐷 = (ω ∖ {∅})
bnj910.11	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj910.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj910.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj910.14	⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj910.15	⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))

Assertion

Ref	Expression
bnj910	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛   𝑓,𝑝,𝑖,𝑛   𝜑,𝑖

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

Proof of Theorem bnj910

Step	Hyp	Ref	Expression
1		bnj910.3	. . . 4 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj910.10	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	bnj970 35497	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷)
4		bnj910.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
5		bnj910.2	. . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		bnj910.12	. . . . 5 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
7		bnj910.14	. . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
8		bnj910.15	. . . . 5 ⊢ (𝜎 ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
9	4, 5, 1, 2, 6, 7, 8	bnj969 35496	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
10		simpr3 1215	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
11	1	bnj1235 35354	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
12	11	3ad2ant1 1151	. . . . 5 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
13	12	adantl 487	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑓 Fn 𝑛)
14		bnj910.13	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
15	14	bnj941 35323	. . . . 5 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
16	15	3impib 1134	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
17	9, 10, 13, 16	syl3anc 1398	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
18		bnj910.4	. . . 4 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
19		bnj910.7	. . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
20	4, 5, 1, 18, 19, 2, 6, 14, 7, 8	bnj944 35488	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜑″)
21		bnj910.5	. . . 4 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
22		bnj910.8	. . . 4 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
23	5, 1, 2, 6, 14, 9	bnj967 35495	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
24	1, 2, 6, 14, 9, 17	bnj966 35494	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
25	5, 1, 21, 22, 6, 14, 23, 24	bnj964 35493	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″)
26	3, 17, 20, 25	bnj951 35326	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
27		bnj910.6	. . . 4 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
28		vex 3454	. . . 4 ⊢ 𝑝 ∈ V
29	1, 18, 21, 27, 28	bnj919 35318	. . 3 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′))
30		bnj910.9	. . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
31	14	bnj918 35317	. . 3 ⊢ 𝐺 ∈ V
32	29, 19, 22, 30, 31	bnj976 35328	. 2 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
33	26, 32	sylibr 237	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  [wsbc 3739   ∖ cdif 3896   ∪ cun 3897  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6354   Fn wfn 6523  ‘cfv 6528  ωcom 7861   ∧ w-bnj17 35237   predc-bnj14 35239   FrSe w-bnj15 35243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-res 5660  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-om 7862  df-bnj17 35238  df-bnj14 35240  df-bnj13 35242  df-bnj15 35244

This theorem is used by:  bnj998  35507