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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege122 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frege116.x | ⊢ 𝑋 ∈ 𝑈 |
frege118.y | ⊢ 𝑌 ∈ 𝑉 |
frege120.a | ⊢ 𝐴 ∈ 𝑊 |
Ref | Expression |
---|---|
frege122 | ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege120.a | . . 3 ⊢ 𝐴 ∈ 𝑊 | |
2 | 1 | frege112 43878 | . 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 43887 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 ∪ cun 3968 class class class wbr 5169 I cid 5596 ◡ccnv 5698 Fun wfun 6566 ‘cfv 6572 t+ctcl 15030 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-frege1 43693 ax-frege2 43694 ax-frege8 43712 ax-frege52a 43760 ax-frege58b 43804 |
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 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-fun 6574 |
This theorem is referenced by: frege123 43889 |
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