Theorem bnj1256

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	bnj1256.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1256.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1256.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1256.4	⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
	bnj1256.5	⊢ 𝐸 = { 𝑥 ∈ 𝐷 ∣ ( 𝑔 ‘ 𝑥 ) ≠ ( ℎ ‘ 𝑥 ) }
	bnj1256.6	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ ( 𝑔 ↾ 𝐷 ) ≠ ( ℎ ↾ 𝐷 ) ) )
	bnj1256.7	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀ 𝑦 ∈ 𝐸 ¬ 𝑦 𝑅 𝑥 ) )
Assertion	bnj1256	⊢ ( 𝜑 → ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 )

Proof

Step	Hyp	Ref	Expression
1		bnj1256.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1256.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1256.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1256.4	⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
5		bnj1256.5	⊢ 𝐸 = { 𝑥 ∈ 𝐷 ∣ ( 𝑔 ‘ 𝑥 ) ≠ ( ℎ ‘ 𝑥 ) }
6		bnj1256.6	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ ( 𝑔 ↾ 𝐷 ) ≠ ( ℎ ↾ 𝐷 ) ) )
7		bnj1256.7	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀ 𝑦 ∈ 𝐸 ¬ 𝑦 𝑅 𝑥 ) )
8		abid	⊢ ( 𝑔 ∈ { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) } ↔ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
9	8	bnj1238	⊢ ( 𝑔 ∈ { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) } → ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 )
10		eqid	⊢ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ = ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
11		eqid	⊢ { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) } = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) }
12	2 3 10 11	bnj1234	⊢ 𝐶 = { 𝑔 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑔 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝑔 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) }
13	9 12	eleq2s	⊢ ( 𝑔 ∈ 𝐶 → ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 )
14	6 13	bnj770	⊢ ( 𝜑 → ∃ 𝑑 ∈ 𝐵 𝑔 Fn 𝑑 )