Theorem bnj1498

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	bnj1498.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1498.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1498.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1498.4	⊢ 𝐹 = ∪ 𝐶
Assertion	bnj1498	⊢ ( 𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		bnj1498.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1498.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1498.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1498.4	⊢ 𝐹 = ∪ 𝐶
5		eliun	⊢ ( 𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃ 𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 )
6	3	bnj1436	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
7	6	bnj1299	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 𝑓 Fn 𝑑 )
8		fndm	⊢ ( 𝑓 Fn 𝑑 → dom 𝑓 = 𝑑 )
9	7 8	bnj31	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 dom 𝑓 = 𝑑 )
10	9	bnj1196	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ( 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) )
11	1	bnj1436	⊢ ( 𝑑 ∈ 𝐵 → ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) )
12	11	simpld	⊢ ( 𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴 )
13	12	anim1i	⊢ ( ( 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑 ) → ( 𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑 ) )
14	10 13	bnj593	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 ( 𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑 ) )
15		sseq1	⊢ ( dom 𝑓 = 𝑑 → ( dom 𝑓 ⊆ 𝐴 ↔ 𝑑 ⊆ 𝐴 ) )
16	15	biimparc	⊢ ( ( 𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑 ) → dom 𝑓 ⊆ 𝐴 )
17	14 16	bnj593	⊢ ( 𝑓 ∈ 𝐶 → ∃ 𝑑 dom 𝑓 ⊆ 𝐴 )
18	17	bnj937	⊢ ( 𝑓 ∈ 𝐶 → dom 𝑓 ⊆ 𝐴 )
19	18	sselda	⊢ ( ( 𝑓 ∈ 𝐶 ∧ 𝑧 ∈ dom 𝑓 ) → 𝑧 ∈ 𝐴 )
20	19	rexlimiva	⊢ ( ∃ 𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 → 𝑧 ∈ 𝐴 )
21	5 20	sylbi	⊢ ( 𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 → 𝑧 ∈ 𝐴 )
22	3	bnj1317	⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑓 𝑤 ∈ 𝐶 )
23	22	bnj1400	⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
24	21 23	eleq2s	⊢ ( 𝑧 ∈ dom ∪ 𝐶 → 𝑧 ∈ 𝐴 )
25	4	dmeqi	⊢ dom 𝐹 = dom ∪ 𝐶
26	24 25	eleq2s	⊢ ( 𝑧 ∈ dom 𝐹 → 𝑧 ∈ 𝐴 )
27	26	ssriv	⊢ dom 𝐹 ⊆ 𝐴
28	27	a1i	⊢ ( 𝑅 FrSe 𝐴 → dom 𝐹 ⊆ 𝐴 )
29	1 2 3	bnj1493	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
30		vsnid	⊢ 𝑥 ∈ { 𝑥 }
31		elun1	⊢ ( 𝑥 ∈ { 𝑥 } → 𝑥 ∈ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
32	30 31	ax-mp	⊢ 𝑥 ∈ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
33		eleq2	⊢ ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → ( 𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
34	32 33	mpbiri	⊢ ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑥 ∈ dom 𝑓 )
35	34	reximi	⊢ ( ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
36	35	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
37	29 36	syl	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
38		eliun	⊢ ( 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
39	38	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓 )
40	37 39	sylibr	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 )
41		nfcv	⊢ Ⅎ 𝑥 𝐴
42	1	bnj1309	⊢ ( 𝑡 ∈ 𝐵 → ∀ 𝑥 𝑡 ∈ 𝐵 )
43	3 42	bnj1307	⊢ ( 𝑡 ∈ 𝐶 → ∀ 𝑥 𝑡 ∈ 𝐶 )
44	43	nfcii	⊢ Ⅎ 𝑥 𝐶
45		nfcv	⊢ Ⅎ 𝑥 dom 𝑓
46	44 45	nfiun	⊢ Ⅎ 𝑥 ∪ 𝑓 ∈ 𝐶 dom 𝑓
47	41 46	dfss3f	⊢ ( 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 )
48	40 47	sylibr	⊢ ( 𝑅 FrSe 𝐴 → 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 )
49	48 23	sseqtrrdi	⊢ ( 𝑅 FrSe 𝐴 → 𝐴 ⊆ dom ∪ 𝐶 )
50	49 25	sseqtrrdi	⊢ ( 𝑅 FrSe 𝐴 → 𝐴 ⊆ dom 𝐹 )
51	28 50	eqssd	⊢ ( 𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴 )