Theorem bnj1321

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	bnj1321.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1321.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1321.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1321.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
Assertion	bnj1321	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏 ) → ∃! 𝑓 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		bnj1321.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1321.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1321.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1321.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		simpr	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏 ) → ∃ 𝑓 𝜏 )
6		simp1	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑅 FrSe 𝐴 )
7	4	simplbi	⊢ ( 𝜏 → 𝑓 ∈ 𝐶 )
8	7	3ad2ant2	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 ∈ 𝐶 )
9		nfab1	⊢ Ⅎ 𝑓 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
10	3 9	nfcxfr	⊢ Ⅎ 𝑓 𝐶
11	10	nfcri	⊢ Ⅎ 𝑓 𝑔 ∈ 𝐶
12		nfv	⊢ Ⅎ 𝑓 dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
13	11 12	nfan	⊢ Ⅎ 𝑓 ( 𝑔 ∈ 𝐶 ∧ dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
14		eleq1w	⊢ ( 𝑓 = 𝑔 → ( 𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶 ) )
15		dmeq	⊢ ( 𝑓 = 𝑔 → dom 𝑓 = dom 𝑔 )
16	15	eqeq1d	⊢ ( 𝑓 = 𝑔 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
17	14 16	anbi12d	⊢ ( 𝑓 = 𝑔 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑔 ∈ 𝐶 ∧ dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) )
18	4 17	syl5bb	⊢ ( 𝑓 = 𝑔 → ( 𝜏 ↔ ( 𝑔 ∈ 𝐶 ∧ dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) )
19	13 18	sbiev	⊢ ( [ 𝑔 / 𝑓 ] 𝜏 ↔ ( 𝑔 ∈ 𝐶 ∧ dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
20	19	simplbi	⊢ ( [ 𝑔 / 𝑓 ] 𝜏 → 𝑔 ∈ 𝐶 )
21	20	3ad2ant3	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑔 ∈ 𝐶 )
22		eqid	⊢ ( dom 𝑓 ∩ dom 𝑔 ) = ( dom 𝑓 ∩ dom 𝑔 )
23	1 2 3 22	bnj1326	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) )
24	6 8 21 23	syl3anc	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) )
25	4	simprbi	⊢ ( 𝜏 → dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
26	25	3ad2ant2	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
27	19	simprbi	⊢ ( [ 𝑔 / 𝑓 ] 𝜏 → dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
28	27	3ad2ant3	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → dom 𝑔 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
29	26 28	eqtr4d	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → dom 𝑓 = dom 𝑔 )
30		bnj1322	⊢ ( dom 𝑓 = dom 𝑔 → ( dom 𝑓 ∩ dom 𝑔 ) = dom 𝑓 )
31	30	reseq2d	⊢ ( dom 𝑓 = dom 𝑔 → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑓 ↾ dom 𝑓 ) )
32	29 31	syl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑓 ↾ dom 𝑓 ) )
33		releq	⊢ ( 𝑧 = 𝑓 → ( Rel 𝑧 ↔ Rel 𝑓 ) )
34	1 2 3	bnj66	⊢ ( 𝑧 ∈ 𝐶 → Rel 𝑧 )
35	33 34	vtoclga	⊢ ( 𝑓 ∈ 𝐶 → Rel 𝑓 )
36		resdm	⊢ ( Rel 𝑓 → ( 𝑓 ↾ dom 𝑓 ) = 𝑓 )
37	8 35 36	3syl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑓 ↾ dom 𝑓 ) = 𝑓 )
38	32 37	eqtrd	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑓 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = 𝑓 )
39		eqeq2	⊢ ( dom 𝑓 = dom 𝑔 → ( ( dom 𝑓 ∩ dom 𝑔 ) = dom 𝑓 ↔ ( dom 𝑓 ∩ dom 𝑔 ) = dom 𝑔 ) )
40	30 39	mpbid	⊢ ( dom 𝑓 = dom 𝑔 → ( dom 𝑓 ∩ dom 𝑔 ) = dom 𝑔 )
41	40	reseq2d	⊢ ( dom 𝑓 = dom 𝑔 → ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ dom 𝑔 ) )
42	29 41	syl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = ( 𝑔 ↾ dom 𝑔 ) )
43	1 2 3	bnj66	⊢ ( 𝑔 ∈ 𝐶 → Rel 𝑔 )
44		resdm	⊢ ( Rel 𝑔 → ( 𝑔 ↾ dom 𝑔 ) = 𝑔 )
45	21 43 44	3syl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑔 ↾ dom 𝑔 ) = 𝑔 )
46	42 45	eqtrd	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → ( 𝑔 ↾ ( dom 𝑓 ∩ dom 𝑔 ) ) = 𝑔 )
47	24 38 46	3eqtr3d	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 = 𝑔 )
48	47	3expib	⊢ ( 𝑅 FrSe 𝐴 → ( ( 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 = 𝑔 ) )
49	48	alrimivv	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑓 ∀ 𝑔 ( ( 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 = 𝑔 ) )
50	49	adantr	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏 ) → ∀ 𝑓 ∀ 𝑔 ( ( 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 = 𝑔 ) )
51		nfv	⊢ Ⅎ 𝑔 𝜏
52	51	eu2	⊢ ( ∃! 𝑓 𝜏 ↔ ( ∃ 𝑓 𝜏 ∧ ∀ 𝑓 ∀ 𝑔 ( ( 𝜏 ∧ [ 𝑔 / 𝑓 ] 𝜏 ) → 𝑓 = 𝑔 ) ) )
53	5 50 52	sylanbrc	⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏 ) → ∃! 𝑓 𝜏 )