Theorem frege123 44439

Description: Lemma for frege124 44440. 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 3435	. . . 4 ⊢ 𝑎 ∈ V
4	1, 2, 3	frege122 44438	. . 3 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
5	4	alrimdv 1936	. 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
6		frege19 44277	. 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 1545   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881   class class class wbr 5073   I cid 5513  ^◡ccnv 5618  Fun wfun 6480  ‘cfv 6486  t+ctcl 14939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-pr 5363  ax-frege1 44243  ax-frege2 44244  ax-frege8 44262  ax-frege52a 44310  ax-frege58b 44354

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ifp 1069  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-fun 6488

This theorem is referenced by:  frege124  44440