Theorem bnj605 35104

Description: Technical lemma. 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
bnj605.5	⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
bnj605.13	⊢ (𝜑″ ↔ [𝑓 / 𝑓]𝜑)
bnj605.14	⊢ (𝜓″ ↔ [𝑓 / 𝑓]𝜓)
bnj605.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj605.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj605.28	⊢ 𝑓 ∈ V
bnj605.31	⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
bnj605.32	⊢ (𝜑″ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj605.33	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj605.37	⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝𝜂)
bnj605.38	⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → 𝜒′)
bnj605.41	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝑓 Fn 𝑛)
bnj605.42	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
bnj605.43	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)

Assertion

Ref	Expression
bnj605	⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))

Distinct variable groups:   𝐴,𝑓,𝑚   𝐴,𝑝,𝑓   𝑅,𝑓,𝑚   𝑅,𝑝   𝜂,𝑓   𝑚,𝑛   𝜑,𝑚   𝜓,𝑚   𝑥,𝑚   𝑛,𝑝   𝜑,𝑝   𝜓,𝑝   𝜃,𝑝   𝑥,𝑝

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑦,𝑖,𝑛)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑦,𝑖,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj605

Step	Hyp	Ref	Expression
1		bnj605.37	. . . . 5 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝𝜂)
2	1	anim1i 622	. . . 4 ⊢ (((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → (∃𝑚∃𝑝𝜂 ∧ 𝜃))
3		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑝𝜃
4	3	19.41 2249	. . . . . 6 ⊢ (∃𝑝(𝜂 ∧ 𝜃) ↔ (∃𝑝𝜂 ∧ 𝜃))
5	4	exbii 1856	. . . . 5 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) ↔ ∃𝑚(∃𝑝𝜂 ∧ 𝜃))
6		bnj605.5	. . . . . . . 8 ⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
7	6	bnj1095 34979	. . . . . . 7 ⊢ (𝜃 → ∀𝑚𝜃)
8	7	nf5i 2159	. . . . . 6 ⊢ Ⅎ𝑚𝜃
9	8	19.41 2249	. . . . 5 ⊢ (∃𝑚(∃𝑝𝜂 ∧ 𝜃) ↔ (∃𝑚∃𝑝𝜂 ∧ 𝜃))
10	5, 9	bitr2i 278	. . . 4 ⊢ ((∃𝑚∃𝑝𝜂 ∧ 𝜃) ↔ ∃𝑚∃𝑝(𝜂 ∧ 𝜃))
11	2, 10	sylib 220	. . 3 ⊢ (((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → ∃𝑚∃𝑝(𝜂 ∧ 𝜃))
12		bnj605.19	. . . . . . . . . 10 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
13	12	bnj1232 35000	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
14		bnj219 34931	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)
15	12, 14	bnj770 34961	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 E 𝑛)
16	13, 15	jca 517	. . . . . . . 8 ⊢ (𝜂 → (𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛))
17	16	anim1i 622	. . . . . . 7 ⊢ ((𝜂 ∧ 𝜃) → ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
18		bnj170 34896	. . . . . . 7 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ↔ ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
19	17, 18	sylibr 236	. . . . . 6 ⊢ ((𝜂 ∧ 𝜃) → (𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛))
20		bnj605.38	. . . . . 6 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → 𝜒′)
21	19, 20	syl 17	. . . . 5 ⊢ ((𝜂 ∧ 𝜃) → 𝜒′)
22		simpl 484	. . . . 5 ⊢ ((𝜂 ∧ 𝜃) → 𝜂)
23	21, 22	jca 517	. . . 4 ⊢ ((𝜂 ∧ 𝜃) → (𝜒′ ∧ 𝜂))
24	23	2eximi 1844	. . 3 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) → ∃𝑚∃𝑝(𝜒′ ∧ 𝜂))
25		bnj248 34898	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) ∧ 𝜂))
26		bnj605.31	. . . . . . . . . . 11 ⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
27		pm3.35 809	. . . . . . . . . . 11 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
28	26, 27	sylan2b 601	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
29		euex 2583	. . . . . . . . . 10 ⊢ (∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → ∃𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
31		bnj605.17	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
32	30, 31	bnj1198 34992	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
33	25, 32	bnj832 34956	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → ∃𝑓𝜏)
34		bnj605.41	. . . . . . . . . . . . 13 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝑓 Fn 𝑛)
35		bnj605.42	. . . . . . . . . . . . 13 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
36		bnj605.43	. . . . . . . . . . . . 13 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)
37	34, 35, 36	3jca 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
38	37	3com23 1133	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂 ∧ 𝜏) → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
39	38	3expia 1128	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (𝜏 → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
40	39	eximdv 1925	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
41	40	ad4ant14 759	. . . . . . . 8 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
42	25, 41	sylbi 219	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
43	33, 42	mpd 15	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
44		bnj432 34914	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) ↔ ((𝜒′ ∧ 𝜂) ∧ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)))
45		biid 263	. . . . . . . 8 ⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛)
46		bnj605.13	. . . . . . . . 9 ⊢ (𝜑″ ↔ [𝑓 / 𝑓]𝜑)
47		sbcid 3742	. . . . . . . . 9 ⊢ ([𝑓 / 𝑓]𝜑 ↔ 𝜑)
48	46, 47	bitri 277	. . . . . . . 8 ⊢ (𝜑″ ↔ 𝜑)
49		bnj605.14	. . . . . . . . 9 ⊢ (𝜓″ ↔ [𝑓 / 𝑓]𝜓)
50		sbcid 3742	. . . . . . . . 9 ⊢ ([𝑓 / 𝑓]𝜓 ↔ 𝜓)
51	49, 50	bitri 277	. . . . . . . 8 ⊢ (𝜓″ ↔ 𝜓)
52	45, 48, 51	3anbi123i 1162	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
53	52	exbii 1856	. . . . . 6 ⊢ (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
54	43, 44, 53	3imtr3i 293	. . . . 5 ⊢ (((𝜒′ ∧ 𝜂) ∧ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
55	54	ex 414	. . . 4 ⊢ ((𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
56	55	exlimivv 1940	. . 3 ⊢ (∃𝑚∃𝑝(𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
57	11, 24, 56	3syl 18	. 2 ⊢ (((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
58	57	3impa 1116	1 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃!weu 2574   ≠ wne 2936  ∀wral 3055  Vcvv 3433  [wsbc 3725  ∅c0 4264  ∪ ciun 4924   class class class wbr 5075   E cep 5520  suc csuc 6316   Fn wfn 6484  ‘cfv 6489  ωcom 7810  1_oc1o 8392   ∧ w-bnj17 34884   predc-bnj14 34886   FrSe w-bnj15 34890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-eprel 5521  df-suc 6320  df-bnj17 34885

This theorem is referenced by: (None)