Metamath Proof Explorer

Theorem satfvsucom

Description: The satisfaction predicate as function over wff codes at a successor of _om . (Contributed by AV, 22-Sep-2023)

		Ref	Expression
	Hypothesis	satfvsucom.s	$⊢ S = M Sat E$
	Assertion	satfvsucom	$⊢ (M \in V \land E \in W \land N \in suc ω) \to S (N) = rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) (N)$

Proof

Step	Hyp	Ref	Expression
1		satfvsucom.s	$⊢ S = M Sat E$
2		satf	$⊢ (M \in V \land E \in W) \to M Sat E = {rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}$
3	2	3adant3	$⊢ (M \in V \land E \in W \land N \in suc ω) \to M Sat E = {rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}$
4	1 3	syl5eq	$⊢ (M \in V \land E \in W \land N \in suc ω) \to S = {rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}$
5	4	fveq1d	$⊢ (M \in V \land E \in W \land N \in suc ω) \to S (N) = ({rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}) (N)$
6		fvres	$⊢ N \in suc ω \to ({rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}) (N) = rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) (N)$
7	6	3ad2ant3	$⊢ (M \in V \land E \in W \land N \in suc ω) \to ({rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) ↾}_{suc ω}) (N) = rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) (N)$
8	5 7	eqtrd	$⊢ (M \in V \land E \in W \land N \in suc ω) \to S (N) = rec ((f \in V ⟼ f \cup \{⟨x, y⟩ \| \exists u \in f (\exists v \in f (x = 1^{st} (u) ⊼_{𝑔} 1^{st} (v) \land y = (M^{ω}) ∖ (2^{nd} (u) \cap 2^{nd} (v))) \lor \exists i \in ω (x = [\forall_{𝑔} i 1^{st} (u)] \land y = \{a \in (M^{ω}) \| \forall z \in M \{⟨i, z⟩\} \cup ({a ↾}_{(ω ∖ \{i\})}) \in 2^{nd} (u)\}))\}), \{⟨x, y⟩ \| \exists i \in ω \exists j \in ω (x = i \in_{𝑔} j \land y = \{a \in (M^{ω}) \| a (i) E a (j)\})\}) (N)$