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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 40411 | . 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 40420 | . 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 2114 ∪ cun 3922 class class class wbr 5052 I cid 5445 ◡ccnv 5540 Fun wfun 6335 ‘cfv 6341 t+ctcl 14330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-frege1 40226 ax-frege2 40227 ax-frege8 40245 ax-frege52a 40293 ax-frege58b 40337 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-fun 6343 |
This theorem is referenced by: frege123 40422 |
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