Theorem frege119

Description: Lemma for frege120 . Proposition 119 of Frege1879 p. 78. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege116.x	⊢ 𝑋 ∈ 𝑈
	frege118.y	⊢ 𝑌 ∈ 𝑉
Assertion	frege119	⊢ ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege116.x	⊢ 𝑋 ∈ 𝑈
2		frege118.y	⊢ 𝑌 ∈ 𝑉
3	1 2	frege118	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) )
4		frege19	⊢ ( ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) ) ) → ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ∀ 𝑎 ( 𝑌 𝑅 𝑎 → 𝑎 = 𝑋 ) → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) → ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) ) )