Theorem frege130

Description: Lemma for frege131 . Proposition 130 of Frege1879 p. 84. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege130.m	⊢ 𝑀 ∈ 𝑈
	frege130.r	⊢ 𝑅 ∈ 𝑉
Assertion	frege130	⊢ ( ( ∀ 𝑏 ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) → ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege130.m	⊢ 𝑀 ∈ 𝑈
2		frege130.r	⊢ 𝑅 ∈ 𝑉
3		vex	⊢ 𝑎 ∈ V
4		vex	⊢ 𝑏 ∈ V
5	3 4 1 2	frege129	⊢ ( Fun ◡ ◡ 𝑅 → ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) )
6	5	alrimdv	⊢ ( Fun ◡ ◡ 𝑅 → ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) )
7	6	alrimiv	⊢ ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) )
8		frege9	⊢ ( ( Fun ◡ ◡ 𝑅 → ∀ 𝑏 ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) ) → ( ( ∀ 𝑏 ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) → ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) )
9	7 8	ax-mp	⊢ ( ( ∀ 𝑏 ( ( ¬ 𝑏 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑏 ) → ∀ 𝑎 ( 𝑏 𝑅 𝑎 → ( ¬ 𝑎 ( t+ ‘ 𝑅 ) 𝑀 → 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) → ( Fun ◡ ◡ 𝑅 → 𝑅 hereditary ( ( ◡ ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) )