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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege121 | Structured version Visualization version GIF version |
Description: Lemma for frege122 40338. Proposition 121 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 |
---|---|
frege121 | ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege116.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
2 | frege118.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
3 | frege120.a | . . 3 ⊢ 𝐴 ∈ 𝑊 | |
4 | 1, 2, 3 | frege120 40336 | . 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
5 | frege20 40181 | . 2 ⊢ ((Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) → ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 class class class wbr 5068 I cid 5461 ◡ccnv 5556 Fun wfun 6351 ‘cfv 6357 t+ctcl 14347 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-frege1 40143 ax-frege2 40144 ax-frege8 40162 ax-frege52a 40210 ax-frege58b 40254 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-fun 6359 |
This theorem is referenced by: frege122 40338 |
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