Theorem bnj151 34874

Description: Technical lemma for bnj153 34877. 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 34873	. . . . . 6 ⊢ 𝜃0
13	12, 6	mpbi 230	. . . . 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 34866	. . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
20	14, 15, 16, 17, 18, 19	bnj149 34872	. . . . . 6 ⊢ 𝜃1
21	20, 14	mpbi 230	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
22		df-eu 2563	. . . . 5 ⊢ (∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ∧ ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
23	13, 21, 22	sylanbrc 583	. . . 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 34871	. . . 4 ⊢ (𝜃′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
27	23, 26	mpbir 231	. . 3 ⊢ 𝜃′
28		sbceq1a 3767	. . . 4 ⊢ (𝑛 = 1o → (𝜃 ↔ [1o / 𝑛]𝜃))
29	28, 25	bitr4di 289	. . 3 ⊢ (𝑛 = 1o → (𝜃 ↔ 𝜃′))
30	27, 29	mpbiri 258	. 2 ⊢ (𝑛 = 1o → 𝜃)
31	30	a1d 25	1 ⊢ (𝑛 = 1o → ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  ∃!weu 2562  ∀wral 3045  [wsbc 3756   ∖ cdif 3914  ∅c0 4299  {csn 4592  ⟨cop 4598  ∪ ciun 4958   class class class wbr 5110   E cep 5540  suc csuc 6337   Fn wfn 6509  ‘cfv 6514  ωcom 7845  1_oc1o 8430   predc-bnj14 34685   FrSe w-bnj15 34689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-1o 8437  df-bnj13 34688  df-bnj15 34690

This theorem is referenced by:  bnj153  34877