Theorem bnj605 30977

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 592	. . . 4 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → (∃𝑚∃𝑝𝜂 ∧ 𝜃))
3		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑝𝜃
4	3	19.41 2103	. . . . . 6 ⊢ (∃𝑝(𝜂 ∧ 𝜃) ↔ (∃𝑝𝜂 ∧ 𝜃))
5	4	exbii 1774	. . . . 5 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) ↔ ∃𝑚(∃𝑝𝜂 ∧ 𝜃))
6		bnj605.5	. . . . . . . 8 ⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
7	6	bnj1095 30852	. . . . . . 7 ⊢ (𝜃 → ∀𝑚𝜃)
8	7	nf5i 2024	. . . . . 6 ⊢ Ⅎ𝑚𝜃
9	8	19.41 2103	. . . . 5 ⊢ (∃𝑚(∃𝑝𝜂 ∧ 𝜃) ↔ (∃𝑚∃𝑝𝜂 ∧ 𝜃))
10	5, 9	bitr2i 265	. . . 4 ⊢ ((∃𝑚∃𝑝𝜂 ∧ 𝜃) ↔ ∃𝑚∃𝑝(𝜂 ∧ 𝜃))
11	2, 10	sylib 208	. . 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 30874	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
14		bnj219 30801	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)
15	12, 14	bnj770 30833	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 E 𝑛)
16	13, 15	jca 554	. . . . . . . 8 ⊢ (𝜂 → (𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛))
17	16	anim1i 592	. . . . . . 7 ⊢ ((𝜂 ∧ 𝜃) → ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
18		bnj170 30764	. . . . . . 7 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ↔ ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
19	17, 18	sylibr 224	. . . . . 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 473	. . . . 5 ⊢ ((𝜂 ∧ 𝜃) → 𝜂)
23	21, 22	jca 554	. . . 4 ⊢ ((𝜂 ∧ 𝜃) → (𝜒′ ∧ 𝜂))
24	23	2eximi 1763	. . 3 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) → ∃𝑚∃𝑝(𝜒′ ∧ 𝜂))
25		bnj248 30766	. . . . . . . 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		pm3.35 611	. . . . . . . . . . 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 492	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
29		euex 2494	. . . . . . . . . 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 30866	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
33	25, 32	bnj832 30828	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → ∃𝑓𝜏)
34		bnj605.41	. . . . . . . . . . . . . 14 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝑓 Fn 𝑛)
35		bnj605.42	. . . . . . . . . . . . . 14 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
36		bnj605.43	. . . . . . . . . . . . . 14 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)
37	34, 35, 36	3jca 1242	. . . . . . . . . . . . 13 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
38	37	3com23 1271	. . . . . . . . . . . 12 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂 ∧ 𝜏) → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
39	38	3expia 1267	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (𝜏 → (𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
40	39	eximdv 1846	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
41	40	adantlr 751	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
42	41	adantlr 751	. . . . . . . 8 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
43	25, 42	sylbi 207	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″)))
44	33, 43	mpd 15	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
45		bnj432 30782	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒′ ∧ 𝜂) ↔ ((𝜒′ ∧ 𝜂) ∧ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)))
46		biid 251	. . . . . . . 8 ⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛)
47		bnj605.13	. . . . . . . . 9 ⊢ (𝜑″ ↔ [𝑓 / 𝑓]𝜑)
48		sbcid 3452	. . . . . . . . 9 ⊢ ([𝑓 / 𝑓]𝜑 ↔ 𝜑)
49	47, 48	bitri 264	. . . . . . . 8 ⊢ (𝜑″ ↔ 𝜑)
50		bnj605.14	. . . . . . . . 9 ⊢ (𝜓″ ↔ [𝑓 / 𝑓]𝜓)
51		sbcid 3452	. . . . . . . . 9 ⊢ ([𝑓 / 𝑓]𝜓 ↔ 𝜓)
52	50, 51	bitri 264	. . . . . . . 8 ⊢ (𝜓″ ↔ 𝜓)
53	46, 49, 52	3anbi123i 1251	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
54	53	exbii 1774	. . . . . 6 ⊢ (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
55	44, 45, 54	3imtr3i 280	. . . . 5 ⊢ (((𝜒′ ∧ 𝜂) ∧ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
56	55	ex 450	. . . 4 ⊢ ((𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
57	56	exlimivv 1860	. . 3 ⊢ (∃𝑚∃𝑝(𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
58	11, 24, 57	3syl 18	. 2 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
59	58	3impa 1259	1 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470   ≠ wne 2794  ∀wral 2912  Vcvv 3200  [wsbc 3435  ∅c0 3915  ∪ ciun 4520   class class class wbr 4653   E cep 5028  suc csuc 5725   Fn wfn 5883  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553   ∧ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-suc 5729  df-bnj17 30753

This theorem is referenced by: (None)