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Metamath Proof Explorer

Theorem oif

Description: The order isomorphism of the well-order R on A is a function. (Contributed by Mario Carneiro, 23-May-2015)

Ref Expression
Hypothesis oicl.1 ⊢ F = OrdIso R A
Assertion oif ⊢ F : dom ⁡ F ⟶ A

Proof

Step Hyp Ref Expression
1 oicl.1 ⊢ F = OrdIso R A
2 eqid ⊢ recs ⁡ h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v = recs ⁡ h ∈ V ⟼ ι 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 ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v = h ∈ V ⟼ ι 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 ⟼ ι s ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y | ∀ r ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y ¬ r R s = recs ⁡ h ∈ V ⟼ ι 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 ⟼ ι 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 ⟼ ι 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 simprd ⊢ R We A ∧ R Se A → F : dom ⁡ F ⟶ A
11 f0 ⊢ ∅ : ∅ ⟶ A
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 12 15 feq12d ⊢ ¬ R We A ∧ R Se A → F : dom ⁡ F ⟶ A ↔ ∅ : ∅ ⟶ A
17 11 16 mpbiri ⊢ ¬ R We A ∧ R Se A → F : dom ⁡ F ⟶ A
18 10 17 pm2.61i ⊢ F : dom ⁡ F ⟶ A