Theorem frege123 43202

Description: Lemma for frege124 43203. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege123.x	⊢ 𝑋 ∈ 𝑈
frege123.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege123	⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

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

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

Proof of Theorem frege123

Step	Hyp	Ref	Expression
1		frege123.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
2		frege123.y	. . . 4 ⊢ 𝑌 ∈ 𝑉
3		vex 3477	. . . 4 ⊢ 𝑎 ∈ V
4	1, 2, 3	frege122 43201	. . 3 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
5	4	alrimdv 1931	. 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
6		frege19 43040	. 2 ⊢ ((Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎))) → ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))))
7	5, 6	ax-mp 5	1 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946   class class class wbr 5148   I cid 5573  ^◡ccnv 5675  Fun wfun 6537  ‘cfv 6543  t+ctcl 14939

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-frege1 43006  ax-frege2 43007  ax-frege8 43025  ax-frege52a 43073  ax-frege58b 43117

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545

This theorem is referenced by:  frege124  43203