Theorem oicl

Description: The order type of the well-order R on A is an ordinal. (Contributed by Mario Carneiro, 23-May-2015) (Revised by Mario Carneiro, 25-Jun-2015)

	Ref	Expression
Hypothesis	oicl.1	⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 )
Assertion	oicl	⊢ Ord dom 𝐹

Proof

Step	Hyp	Ref	Expression
1		oicl.1	⊢ 𝐹 = OrdIso ( 𝑅 , 𝐴 )
2		eqid	⊢ recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) ) = recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) )
3		eqid	⊢ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } = { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 }
4		eqid	⊢ ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) = ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) )
5	2 3 4	ordtypecbv	⊢ recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) = recs ( ( ℎ ∈ V ↦ ( ℩ 𝑣 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ∀ 𝑢 ∈ { 𝑤 ∈ 𝐴 ∣ ∀ 𝑗 ∈ ran ℎ 𝑗 𝑅 𝑤 } ¬ 𝑢 𝑅 𝑣 ) ) )
6		eqid	⊢ { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) “ 𝑥 ) 𝑧 𝑅 𝑡 } = { 𝑥 ∈ On ∣ ∃ 𝑡 ∈ 𝐴 ∀ 𝑧 ∈ ( recs ( ( 𝑓 ∈ V ↦ ( ℩ 𝑠 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ∀ 𝑟 ∈ { 𝑦 ∈ 𝐴 ∣ ∀ 𝑖 ∈ ran 𝑓 𝑖 𝑅 𝑦 } ¬ 𝑟 𝑅 𝑠 ) ) ) “ 𝑥 ) 𝑧 𝑅 𝑡 }
7		simpl	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 We 𝐴 )
8		simpr	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 )
9	5 3 4 6 1 7 8	ordtypelem5	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( Ord dom 𝐹 ∧ 𝐹 : dom 𝐹 ⟶ 𝐴 ) )
10	9	simpld	⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → Ord dom 𝐹 )
11		ord0	⊢ Ord ∅
12	1	oi0	⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 = ∅ )
13	12	dmeqd	⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = dom ∅ )
14		dm0	⊢ dom ∅ = ∅
15	13 14	eqtrdi	⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = ∅ )
16		ordeq	⊢ ( dom 𝐹 = ∅ → ( Ord dom 𝐹 ↔ Ord ∅ ) )
17	15 16	syl	⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( Ord dom 𝐹 ↔ Ord ∅ ) )
18	11 17	mpbiri	⊢ ( ¬ ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → Ord dom 𝐹 )
19	10 18	pm2.61i	⊢ Ord dom 𝐹