Theorem bnj151 35373

Description: Technical lemma for bnj153 35376. 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
bnj151.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj151.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj151.3	⊢ 𝐷 = (ω ∖ {∅})
bnj151.4	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj151.5	⊢ (𝜏 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜃))
bnj151.6	⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj151.7	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
bnj151.8	⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
bnj151.9	⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)
bnj151.10	⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
bnj151.11	⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
bnj151.12	⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)
bnj151.13	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj151.14	⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
bnj151.15	⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
bnj151.16	⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
bnj151.17	⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
bnj151.18	⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
bnj151.19	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj151.20	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)

Assertion

Ref	Expression
bnj151	⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥   𝐴,𝑛,𝑓,𝑥   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑔,𝑥   𝑅,𝑛   𝑓,𝜁₁   𝑔,𝜁₀   𝑖,𝑛,𝑦   𝑚,𝑛

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜓(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜃(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜏(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜁(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝐴(𝑦, 𝑖, 𝑚)   𝐷(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝑅(𝑦, 𝑖, 𝑚)   𝐹(𝑥, 𝑔, 𝑚, 𝑛)   𝜑′(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜓′(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜃′(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜁′(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜑″(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜓″(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜁″(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜃₀(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜁₀(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜑₁(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜓₁(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜃₁(𝑥, 𝑦, 𝑓, 𝑔, 𝑖, 𝑚, 𝑛)   𝜁₁(𝑥, 𝑦, 𝑔, 𝑖, 𝑚, 𝑛)

Proof of Theorem bnj151

Step	Hyp	Ref	Expression
1		bnj151.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
2		bnj151.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj151.6	. . . . . . 7 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
4		bnj151.7	. . . . . . 7 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
5		bnj151.8	. . . . . . 7 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
6		bnj151.10	. . . . . . 7 ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
7		bnj151.12	. . . . . . 7 ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)
8		bnj151.13	. . . . . . 7 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
9		bnj151.14	. . . . . . 7 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
10		bnj151.15	. . . . . . 7 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
11		bnj151.16	. . . . . . 7 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj150 35372	. . . . . 6 ⊢ 𝜃0
13	12, 6	mpbi 233	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
14		bnj151.11	. . . . . . 7 ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
15		bnj151.17	. . . . . . 7 ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
16		bnj151.18	. . . . . . 7 ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
17		bnj151.19	. . . . . . 7 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
18		bnj151.20	. . . . . . 7 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
19	1, 4	bnj118 35365	. . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
20	14, 15, 16, 17, 18, 19	bnj149 35371	. . . . . 6 ⊢ 𝜃1
21	20, 14	mpbi 233	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
22		df-eu 2596	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ∧ ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
23	13, 21, 22	sylanbrc 595	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
24		bnj151.4	. . . . 5 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
25		bnj151.9	. . . . 5 ⊢ (𝜃′ ↔ [1o / 𝑛]𝜃)
26	24, 4, 5, 25	bnj130 35370	. . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
27	23, 26	mpbir 234	. . 3 ⊢ 𝜃′
28		sbceq1a 3753	. . . 4 ⊢ (𝑛 = 1o → (𝜃 ↔ [1o / 𝑛]𝜃))
29	28, 25	bitr4di 292	. . 3 ⊢ (𝑛 = 1o → (𝜃 ↔ 𝜃′))
30	27, 29	mpbiri 261	. 2 ⊢ (𝑛 = 1o → 𝜃)
31	30	a1d 26	1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2564  ∃!weu 2595  ∀wral 3078  [wsbc 3742   ∖ cdif 3899  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   E cep 5558  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865  1_oc1o 8451   predc-bnj14 35185   FrSe w-bnj15 35189

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8458  df-bnj13 35188  df-bnj15 35190

This theorem is used by:  bnj153  35376