Theorem bnj1388

Description: Technical lemma for bnj60 . 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.)

	Ref	Expression
Hypotheses	bnj1388.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1388.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1388.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1388.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1388.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1388.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1388.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1388.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
Assertion	bnj1388	⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )

Proof

Step	Hyp	Ref	Expression
1		bnj1388.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1388.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1388.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1388.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1388.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1388.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1388.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1388.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		nfv	⊢ Ⅎ 𝑦 𝜓
10		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ 𝐷
11		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥
12	9 10 11	nf3an	⊢ Ⅎ 𝑦 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
13	7 12	nfxfr	⊢ Ⅎ 𝑦 𝜒
14		bnj1152	⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) )
15	14	simplbi	⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → 𝑦 ∈ 𝐴 )
16	15	adantl	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦 ∈ 𝐴 )
17	14	biimpi	⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) )
18	17	adantl	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦 ∈ 𝐴 ∧ 𝑦 𝑅 𝑥 ) )
19	18	simprd	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑦 𝑅 𝑥 )
20	7	simp3bi	⊢ ( 𝜒 → ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
21	20	adantr	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
22		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) )
23		con2b	⊢ ( ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) )
24	23	albii	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐷 → ¬ 𝑦 𝑅 𝑥 ) ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) )
25	22 24	bitri	⊢ ( ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) )
26		sp	⊢ ( ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) → ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) )
27	26	impcom	⊢ ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ( 𝑦 𝑅 𝑥 → ¬ 𝑦 ∈ 𝐷 ) ) → ¬ 𝑦 ∈ 𝐷 )
28	25 27	sylan2b	⊢ ( ( 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) → ¬ 𝑦 ∈ 𝐷 )
29	19 21 28	syl2anc	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ 𝑦 ∈ 𝐷 )
30	5	eleq2i	⊢ ( 𝑦 ∈ 𝐷 ↔ 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } )
31		nfcv	⊢ Ⅎ 𝑥 𝑦
32		nfcv	⊢ Ⅎ 𝑥 𝐴
33		nfsbc1v	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜏
34	8 33	nfxfr	⊢ Ⅎ 𝑥 𝜏′
35	34	nfex	⊢ Ⅎ 𝑥 ∃ 𝑓 𝜏′
36	35	nfn	⊢ Ⅎ 𝑥 ¬ ∃ 𝑓 𝜏′
37		sbceq1a	⊢ ( 𝑥 = 𝑦 → ( 𝜏 ↔ [ 𝑦 / 𝑥 ] 𝜏 ) )
38	37 8	bitr4di	⊢ ( 𝑥 = 𝑦 → ( 𝜏 ↔ 𝜏′ ) )
39	38	exbidv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 𝜏′ ) )
40	39	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ ∃ 𝑓 𝜏 ↔ ¬ ∃ 𝑓 𝜏′ ) )
41	31 32 36 40	elrabf	⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } ↔ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
42	30 41	bitri	⊢ ( 𝑦 ∈ 𝐷 ↔ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
43	29 42	sylnib	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ¬ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
44		iman	⊢ ( ( 𝑦 ∈ 𝐴 → ∃ 𝑓 𝜏′ ) ↔ ¬ ( 𝑦 ∈ 𝐴 ∧ ¬ ∃ 𝑓 𝜏′ ) )
45	43 44	sylibr	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑦 ∈ 𝐴 → ∃ 𝑓 𝜏′ ) )
46	16 45	mpd	⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 𝜏′ )
47	46	ex	⊢ ( 𝜒 → ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∃ 𝑓 𝜏′ ) )
48	13 47	ralrimi	⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ )