Theorem frege123 43977

Description: Lemma for frege124 43978. 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 3468	. . . 4 ⊢ 𝑎 ∈ V
4	1, 2, 3	frege122 43976	. . 3 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
5	4	alrimdv 1929	. 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
6		frege19 43815	. 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 2109  Vcvv 3464   ∪ cun 3929   class class class wbr 5124   I cid 5552  ^◡ccnv 5658  Fun wfun 6530  ‘cfv 6536  t+ctcl 15009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-frege1 43781  ax-frege2 43782  ax-frege8 43800  ax-frege52a 43848  ax-frege58b 43892

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-fun 6538

This theorem is referenced by:  frege124  43978