Theorem bnj607 30240

Description: Technical lemma for bnj852 30245. 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
bnj607.5	⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
bnj607.13	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj607.14	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj607.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj607.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj607.28	⊢ 𝐺 ∈ V
bnj607.31	⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
bnj607.32	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj607.33	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj607.37	⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝𝜂)
bnj607.38	⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → 𝜒′)
bnj607.41	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
bnj607.42	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
bnj607.43	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)
bnj607.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj607.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj607.400	⊢ (𝜑0 ↔ [ℎ / 𝑓]𝜑)
bnj607.401	⊢ (𝜓0 ↔ [ℎ / 𝑓]𝜓)
bnj607.300	⊢ (𝜑1 ↔ [𝐺 / ℎ]𝜑0)
bnj607.301	⊢ (𝜓1 ↔ [𝐺 / ℎ]𝜓0)

Assertion

Ref	Expression
bnj607	⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))

Distinct variable groups:   𝐴,𝑓,ℎ   𝐴,𝑚,𝑓   𝐴,𝑝,𝑓   ℎ,𝐺,𝑖,𝑦   𝑅,𝑓,ℎ   𝑅,𝑚   𝑅,𝑝   𝜂,𝑓   𝑓,𝑖,𝑦   𝑓,𝑛,ℎ   𝑥,𝑓,ℎ   𝜑,ℎ   𝜓,ℎ   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,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜃(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛)   𝜏(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,ℎ,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑦,𝑖,𝑛)   𝐷(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑦,𝑖,𝑛)   𝐺(𝑥,𝑓,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜑₀(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜓₀(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜑₁(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)   𝜓₁(𝑥,𝑦,𝑓,ℎ,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj607

Step	Hyp	Ref	Expression
1		bnj607.37	. . . . 5 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝𝜂)
2	1	anim1i 590	. . . 4 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → (∃𝑚∃𝑝𝜂 ∧ 𝜃))
3		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑝𝜃
4	3	19.41 2090	. . . . . 6 ⊢ (∃𝑝(𝜂 ∧ 𝜃) ↔ (∃𝑝𝜂 ∧ 𝜃))
5	4	exbii 1764	. . . . 5 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) ↔ ∃𝑚(∃𝑝𝜂 ∧ 𝜃))
6		bnj607.5	. . . . . . . 8 ⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
7	6	bnj1095 30106	. . . . . . 7 ⊢ (𝜃 → ∀𝑚𝜃)
8	7	nf5i 2011	. . . . . 6 ⊢ Ⅎ𝑚𝜃
9	8	19.41 2090	. . . . 5 ⊢ (∃𝑚(∃𝑝𝜂 ∧ 𝜃) ↔ (∃𝑚∃𝑝𝜂 ∧ 𝜃))
10	5, 9	bitr2i 264	. . . 4 ⊢ ((∃𝑚∃𝑝𝜂 ∧ 𝜃) ↔ ∃𝑚∃𝑝(𝜂 ∧ 𝜃))
11	2, 10	sylib 207	. . 3 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → ∃𝑚∃𝑝(𝜂 ∧ 𝜃))
12		bnj607.19	. . . . . . . . . 10 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
13	12	bnj1232 30128	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
14		bnj219 30055	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑚 → 𝑚 E 𝑛)
15	12, 14	bnj770 30087	. . . . . . . . 9 ⊢ (𝜂 → 𝑚 E 𝑛)
16	13, 15	jca 553	. . . . . . . 8 ⊢ (𝜂 → (𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛))
17	16	anim1i 590	. . . . . . 7 ⊢ ((𝜂 ∧ 𝜃) → ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
18		bnj170 30017	. . . . . . 7 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ↔ ((𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) ∧ 𝜃))
19	17, 18	sylibr 223	. . . . . 6 ⊢ ((𝜂 ∧ 𝜃) → (𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛))
20		bnj607.38	. . . . . 6 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → 𝜒′)
21	19, 20	syl 17	. . . . 5 ⊢ ((𝜂 ∧ 𝜃) → 𝜒′)
22		simpl 472	. . . . 5 ⊢ ((𝜂 ∧ 𝜃) → 𝜂)
23	21, 22	jca 553	. . . 4 ⊢ ((𝜂 ∧ 𝜃) → (𝜒′ ∧ 𝜂))
24	23	2eximi 1753	. . 3 ⊢ (∃𝑚∃𝑝(𝜂 ∧ 𝜃) → ∃𝑚∃𝑝(𝜒′ ∧ 𝜂))
25		bnj607.31	. . . . . . . . . . . 12 ⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
26	25	biimpi 205	. . . . . . . . . . 11 ⊢ (𝜒′ → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
27		euex 2482	. . . . . . . . . . 11 ⊢ (∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → ∃𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
28	26, 27	syl6 34	. . . . . . . . . 10 ⊢ (𝜒′ → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
29	28	impcom 445	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
30		bnj607.17	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
31	29, 30	bnj1198 30120	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝜒′) → ∃𝑓𝜏)
32	31	adantrr 749	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜒′ ∧ 𝜂)) → ∃𝑓𝜏)
33		id 22	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂))
34	33	3com23 1263	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂 ∧ 𝜏) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂))
35	34	3expia 1259	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (𝜏 → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂)))
36	35	eximdv 1833	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜂) → (∃𝑓𝜏 → ∃𝑓(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂)))
37	36	ad2ant2rl 781	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜒′ ∧ 𝜂)) → (∃𝑓𝜏 → ∃𝑓(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂)))
38	32, 37	mpd 15	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜒′ ∧ 𝜂)) → ∃𝑓(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂))
39		bnj607.41	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
40		bnj607.42	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
41		bnj607.43	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)
42	39, 40, 41	3jca 1235	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
43	42	eximi 1752	. . . . . 6 ⊢ (∃𝑓(𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → ∃𝑓(𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″))
44		nfe1 2014	. . . . . . 7 ⊢ Ⅎ𝑓∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
45		bnj607.28	. . . . . . . . 9 ⊢ 𝐺 ∈ V
46		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎℎ𝐺
47		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎℎ 𝐺 Fn 𝑛
48		bnj607.300	. . . . . . . . . . . 12 ⊢ (𝜑1 ↔ [𝐺 / ℎ]𝜑0)
49		nfsbc1v 3422	. . . . . . . . . . . 12 ⊢ Ⅎℎ[𝐺 / ℎ]𝜑0
50	48, 49	nfxfr 1771	. . . . . . . . . . 11 ⊢ Ⅎℎ𝜑1
51		bnj607.301	. . . . . . . . . . . 12 ⊢ (𝜓1 ↔ [𝐺 / ℎ]𝜓0)
52		nfsbc1v 3422	. . . . . . . . . . . 12 ⊢ Ⅎℎ[𝐺 / ℎ]𝜓0
53	51, 52	nfxfr 1771	. . . . . . . . . . 11 ⊢ Ⅎℎ𝜓1
54	47, 50, 53	nf3an 1819	. . . . . . . . . 10 ⊢ Ⅎℎ(𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1)
55		fneq1 5893	. . . . . . . . . . 11 ⊢ (ℎ = 𝐺 → (ℎ Fn 𝑛 ↔ 𝐺 Fn 𝑛))
56		sbceq1a 3413	. . . . . . . . . . . 12 ⊢ (ℎ = 𝐺 → (𝜑0 ↔ [𝐺 / ℎ]𝜑0))
57	56, 48	syl6bbr 277	. . . . . . . . . . 11 ⊢ (ℎ = 𝐺 → (𝜑0 ↔ 𝜑1))
58		sbceq1a 3413	. . . . . . . . . . . 12 ⊢ (ℎ = 𝐺 → (𝜓0 ↔ [𝐺 / ℎ]𝜓0))
59	58, 51	syl6bbr 277	. . . . . . . . . . 11 ⊢ (ℎ = 𝐺 → (𝜓0 ↔ 𝜓1))
60	55, 57, 59	3anbi123d 1391	. . . . . . . . . 10 ⊢ (ℎ = 𝐺 → ((ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0) ↔ (𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1)))
61	46, 54, 60	spcegf 3262	. . . . . . . . 9 ⊢ (𝐺 ∈ V → ((𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1) → ∃ℎ(ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0)))
62	45, 61	ax-mp 5	. . . . . . . 8 ⊢ ((𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1) → ∃ℎ(ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0))
63		bnj607.32	. . . . . . . . . . . 12 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
64		bnj607.400	. . . . . . . . . . . . . 14 ⊢ (𝜑0 ↔ [ℎ / 𝑓]𝜑)
65		bnj607.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
66	64, 65	bnj154 30202	. . . . . . . . . . . . 13 ⊢ (𝜑0 ↔ (ℎ‘∅) = pred(𝑥, 𝐴, 𝑅))
67	66, 48, 45	bnj526 30212	. . . . . . . . . . . 12 ⊢ (𝜑1 ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
68	63, 67	bitr4i 266	. . . . . . . . . . 11 ⊢ (𝜑″ ↔ 𝜑1)
69		bnj607.33	. . . . . . . . . . . 12 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
70		bnj607.2	. . . . . . . . . . . . . 14 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
71		bnj607.401	. . . . . . . . . . . . . 14 ⊢ (𝜓0 ↔ [ℎ / 𝑓]𝜓)
72		vex 3176	. . . . . . . . . . . . . 14 ⊢ ℎ ∈ V
73	70, 71, 72	bnj540 30216	. . . . . . . . . . . . 13 ⊢ (𝜓0 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (ℎ‘suc 𝑖) = ∪ 𝑦 ∈ (ℎ‘𝑖) pred(𝑦, 𝐴, 𝑅)))
74	73, 51, 45	bnj540 30216	. . . . . . . . . . . 12 ⊢ (𝜓1 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
75	69, 74	bitr4i 266	. . . . . . . . . . 11 ⊢ (𝜓″ ↔ 𝜓1)
76	68, 75	anbi12i 729	. . . . . . . . . 10 ⊢ ((𝜑″ ∧ 𝜓″) ↔ (𝜑1 ∧ 𝜓1))
77	76	anbi2i 726	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝑛 ∧ (𝜑″ ∧ 𝜓″)) ↔ (𝐺 Fn 𝑛 ∧ (𝜑1 ∧ 𝜓1)))
78		3anass 1035	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ (𝐺 Fn 𝑛 ∧ (𝜑″ ∧ 𝜓″)))
79		3anass 1035	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1) ↔ (𝐺 Fn 𝑛 ∧ (𝜑1 ∧ 𝜓1)))
80	77, 78, 79	3bitr4i 291	. . . . . . . 8 ⊢ ((𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) ↔ (𝐺 Fn 𝑛 ∧ 𝜑1 ∧ 𝜓1))
81		nfv 1830	. . . . . . . . 9 ⊢ Ⅎℎ(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
82		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑓 ℎ Fn 𝑛
83		nfsbc1v 3422	. . . . . . . . . . 11 ⊢ Ⅎ𝑓[ℎ / 𝑓]𝜑
84	64, 83	nfxfr 1771	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝜑0
85		nfsbc1v 3422	. . . . . . . . . . 11 ⊢ Ⅎ𝑓[ℎ / 𝑓]𝜓
86	71, 85	nfxfr 1771	. . . . . . . . . 10 ⊢ Ⅎ𝑓𝜓0
87	82, 84, 86	nf3an 1819	. . . . . . . . 9 ⊢ Ⅎ𝑓(ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0)
88		fneq1 5893	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑓 Fn 𝑛 ↔ ℎ Fn 𝑛))
89		sbceq1a 3413	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝜑 ↔ [ℎ / 𝑓]𝜑))
90	89, 64	syl6bbr 277	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝜑 ↔ 𝜑0))
91		sbceq1a 3413	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝜓 ↔ [ℎ / 𝑓]𝜓))
92	91, 71	syl6bbr 277	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝜓 ↔ 𝜓0))
93	88, 90, 92	3anbi123d 1391	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0)))
94	81, 87, 93	cbvex 2260	. . . . . . . 8 ⊢ (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃ℎ(ℎ Fn 𝑛 ∧ 𝜑0 ∧ 𝜓0))
95	62, 80, 94	3imtr4i 280	. . . . . . 7 ⊢ ((𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
96	44, 95	exlimi 2073	. . . . . 6 ⊢ (∃𝑓(𝐺 Fn 𝑛 ∧ 𝜑″ ∧ 𝜓″) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
97	38, 43, 96	3syl 18	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝜒′ ∧ 𝜂)) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
98	97	expcom 450	. . . 4 ⊢ ((𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
99	98	exlimivv 1847	. . 3 ⊢ (∃𝑚∃𝑝(𝜒′ ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
100	11, 24, 99	3syl 18	. 2 ⊢ (((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
101	100	3impa 1251	1 ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∀wral 2896  Vcvv 3173  [wsbc 3402  ∅c0 3874  ∪ ciun 4455   class class class wbr 4583   E cep 4947  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  1_𝑜c1o 7440   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-eprel 4949  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006

This theorem is referenced by:  bnj600  30243