Theorem frege122 43949

Description: If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege122	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))

Proof of Theorem frege122

Step	Hyp	Ref	Expression
1		frege120.a	. . 3 ⊢ 𝐴 ∈ 𝑊
2	1	frege112 43939	. 2 ⊢ (𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴)
3		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
4		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
5	3, 4, 1	frege121 43948	. 2 ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))
6	2, 5	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   class class class wbr 5166   I cid 5592  ^◡ccnv 5699  Fun wfun 6569  ‘cfv 6575  t+ctcl 15036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege1 43754  ax-frege2 43755  ax-frege8 43773  ax-frege52a 43821  ax-frege58b 43865

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6577

This theorem is referenced by:  frege123  43950