Theorem frege123

Description: Lemma for frege124 . Proposition 123 of Frege1879 p. 79. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	⊢ 𝑋 ∈ 𝑈
	frege123.y	⊢ 𝑌 ∈ 𝑉
Assertion	frege123	⊢ ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege123.x	⊢ 𝑋 ∈ 𝑈
2		frege123.y	⊢ 𝑌 ∈ 𝑉
3		vex	⊢ 𝑎 ∈ V
4	1 2 3	frege122	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) )
5	4	alrimdv	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) )
6		frege19	⊢ ( ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) ) ) → ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑎 ) → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 ( t+ ‘ 𝑅 ) 𝑀 → 𝑋 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑀 ) ) ) )