Theorem bnj1373

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	bnj1373.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1373.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1373.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1373.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1373.5	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
Assertion	bnj1373	⊢ ( 𝜏′ ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1373.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1373.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1373.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1373.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1373.5	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
6	1	bnj1309	⊢ ( 𝑓 ∈ 𝐵 → ∀ 𝑥 𝑓 ∈ 𝐵 )
7	3 6	bnj1307	⊢ ( 𝑓 ∈ 𝐶 → ∀ 𝑥 𝑓 ∈ 𝐶 )
8	7	bnj1351	⊢ ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∀ 𝑥 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
9	8	nf5i	⊢ Ⅎ 𝑥 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
10		sneq	⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } )
11		bnj1318	⊢ ( 𝑥 = 𝑦 → trCl ( 𝑥 , 𝐴 , 𝑅 ) = trCl ( 𝑦 , 𝐴 , 𝑅 ) )
12	10 11	uneq12d	⊢ ( 𝑥 = 𝑦 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
13	12	eqeq2d	⊢ ( 𝑥 = 𝑦 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
14	13	anbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
15	4 14	syl5bb	⊢ ( 𝑥 = 𝑦 → ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
16	9 15	sbciegf	⊢ ( 𝑦 ∈ V → ( [ 𝑦 / 𝑥 ] 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
17	16	elv	⊢ ( [ 𝑦 / 𝑥 ] 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
18	5 17	bitri	⊢ ( 𝜏′ ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )