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| Mirrors > Home > MPE Home > Th. List > madecut | Structured version Visualization version GIF version | ||
| Description: Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024.) |
| Ref | Expression |
|---|---|
| madecut | ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simplr 769 | . . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 <<s 𝑅) | |
| 2 | ssltex1 27831 | . . . . 5 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ V) |
| 4 | simprl 771 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ⊆ ( O ‘𝐴)) | |
| 5 | 3, 4 | elpwd 4606 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ 𝒫 ( O ‘𝐴)) |
| 6 | ssltex2 27832 | . . . . 5 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V) | |
| 7 | 1, 6 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ∈ V) |
| 8 | simprr 773 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ⊆ ( O ‘𝐴)) | |
| 9 | 7, 8 | elpwd 4606 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ∈ 𝒫 ( O ‘𝐴)) |
| 10 | eqidd 2738 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) = (𝐿 |s 𝑅)) | |
| 11 | breq1 5146 | . . . . 5 ⊢ (𝑙 = 𝐿 → (𝑙 <<s 𝑟 ↔ 𝐿 <<s 𝑟)) | |
| 12 | oveq1 7438 | . . . . . 6 ⊢ (𝑙 = 𝐿 → (𝑙 |s 𝑟) = (𝐿 |s 𝑟)) | |
| 13 | 12 | eqeq1d 2739 | . . . . 5 ⊢ (𝑙 = 𝐿 → ((𝑙 |s 𝑟) = (𝐿 |s 𝑅) ↔ (𝐿 |s 𝑟) = (𝐿 |s 𝑅))) |
| 14 | 11, 13 | anbi12d 632 | . . . 4 ⊢ (𝑙 = 𝐿 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅)) ↔ (𝐿 <<s 𝑟 ∧ (𝐿 |s 𝑟) = (𝐿 |s 𝑅)))) |
| 15 | breq2 5147 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝐿 <<s 𝑟 ↔ 𝐿 <<s 𝑅)) | |
| 16 | oveq2 7439 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (𝐿 |s 𝑟) = (𝐿 |s 𝑅)) | |
| 17 | 16 | eqeq1d 2739 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((𝐿 |s 𝑟) = (𝐿 |s 𝑅) ↔ (𝐿 |s 𝑅) = (𝐿 |s 𝑅))) |
| 18 | 15, 17 | anbi12d 632 | . . . 4 ⊢ (𝑟 = 𝑅 → ((𝐿 <<s 𝑟 ∧ (𝐿 |s 𝑟) = (𝐿 |s 𝑅)) ↔ (𝐿 <<s 𝑅 ∧ (𝐿 |s 𝑅) = (𝐿 |s 𝑅)))) |
| 19 | 14, 18 | rspc2ev 3635 | . . 3 ⊢ ((𝐿 ∈ 𝒫 ( O ‘𝐴) ∧ 𝑅 ∈ 𝒫 ( O ‘𝐴) ∧ (𝐿 <<s 𝑅 ∧ (𝐿 |s 𝑅) = (𝐿 |s 𝑅))) → ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅))) |
| 20 | 5, 9, 1, 10, 19 | syl112anc 1376 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅))) |
| 21 | elmade2 27907 | . . 3 ⊢ (𝐴 ∈ On → ((𝐿 |s 𝑅) ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅)))) | |
| 22 | 21 | ad2antrr 726 | . 2 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → ((𝐿 |s 𝑅) ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅)))) |
| 23 | 20, 22 | mpbird 257 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 Oncon0 6384 ‘cfv 6561 (class class class)co 7431 <<s csslt 27825 |s cscut 27827 M cmade 27881 O cold 27882 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 |
| This theorem is referenced by: madebday 27938 |
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