Theorem ordtypelem1

Description: Lemma for ordtype . (Contributed by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Hypotheses	ordtypelem.1	⊢ 𝐹 = recs ( 𝐺 )
	ordtypelem.2	⊢ 𝐶 = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 }
	ordtypelem.3	⊢ 𝐺 = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑅 𝑣 ) )
	ordtypelem.5	⊢ 𝑇 = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 }
	ordtypelem.6	⊢ 𝑂 = OrdIso ( 𝑅 , 𝐴 )
	ordtypelem.7	⊢ ( 𝜑 → 𝑅 We 𝐴 )
	ordtypelem.8	⊢ ( 𝜑 → 𝑅 Se 𝐴 )
Assertion	ordtypelem1	⊢ ( 𝜑 → 𝑂 = ( 𝐹 ↾ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		ordtypelem.1	⊢ 𝐹 = recs ( 𝐺 )
2		ordtypelem.2	⊢ 𝐶 = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 }
3		ordtypelem.3	⊢ 𝐺 = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ 𝐶 ∀ 𝑢 ∈ 𝐶 ¬ 𝑢 𝑅 𝑣 ) )
4		ordtypelem.5	⊢ 𝑇 = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 }
5		ordtypelem.6	⊢ 𝑂 = OrdIso ( 𝑅 , 𝐴 )
6		ordtypelem.7	⊢ ( 𝜑 → 𝑅 We 𝐴 )
7		ordtypelem.8	⊢ ( 𝜑 → 𝑅 Se 𝐴 )
8		iftrue	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → if ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) , ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) , ∅ ) = ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) )
9	6 7 8	syl2anc	⊢ ( 𝜑 → if ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) , ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) , ∅ ) = ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) )
10	2 3 1	dfoi	⊢ OrdIso ( 𝑅 , 𝐴 ) = if ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) , ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) , ∅ )
11	5 10	eqtri	⊢ 𝑂 = if ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) , ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } ) , ∅ )
12	4	reseq2i	⊢ ( 𝐹 ↾ 𝑇 ) = ( 𝐹 ↾ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( 𝐹 “ 𝑥 ) 𝑧 𝑅 𝑡 } )
13	9 11 12	3eqtr4g	⊢ ( 𝜑 → 𝑂 = ( 𝐹 ↾ 𝑇 ) )