Theorem frege119 43994

Description: Lemma for frege120 43995. Proposition 119 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege119	⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))

Distinct variable groups:   𝑅,𝑎   𝑋,𝑎   𝑌,𝑎

Allowed substitution hints:   𝐴(𝑎)   𝑈(𝑎)   𝑉(𝑎)

Proof of Theorem frege119

Step	Hyp	Ref	Expression
1		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
3	1, 2	frege118 43993	. 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
4		frege19 43836	. 2 ⊢ ((Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))))
5	3, 4	ax-mp 5	1 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2110   class class class wbr 5089  ^◡ccnv 5613  Fun wfun 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-frege1 43802  ax-frege2 43803  ax-frege8 43821  ax-frege52a 43869  ax-frege58b 43913

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-fun 6479

This theorem is referenced by:  frege120  43995