Theorem frege122 39118

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 39108	. 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 39117	. 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 1656   ∈ wcel 2164   ∪ cun 3796   class class class wbr 4875   I cid 5251  ^◡ccnv 5345  Fun wfun 6121  ‘cfv 6127  t+ctcl 14110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege52a 38990  ax-frege58b 39034

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-fun 6129

This theorem is referenced by:  frege123  39119