Metamath Proof Explorer

Theorem oiiniseg

Description: ran F is an initial segment of A under the well-order R . (Contributed by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypothesis oicl.1 ⊢ F = OrdIso R A
Assertion oiiniseg ⊢ R We A ∧ R Se A ∧ N ∈ A ∧ M ∈ dom ⁡ F → F ⁡ M R N ∨ N ∈ ran ⁡ F

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 ordtypelem7 ⊢ R We A ∧ R Se A ∧ N ∈ A ∧ M ∈ dom ⁡ F → F ⁡ M R N ∨ N ∈ ran ⁡ F
10 9 anasss ⊢ R We A ∧ R Se A ∧ N ∈ A ∧ M ∈ dom ⁡ F → F ⁡ M R N ∨ N ∈ ran ⁡ F