Theorem madecut 27851

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.)

Assertion

Ref	Expression
madecut	⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴))

Proof of Theorem madecut

Dummy variables 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 <<s 𝑅)
2		ssltex1 27755	. . . . 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 770	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ⊆ ( O ‘𝐴))
5	3, 4	elpwd 4586	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ 𝒫 ( O ‘𝐴))
6		ssltex2 27756	. . . . 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 772	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ⊆ ( O ‘𝐴))
9	7, 8	elpwd 4586	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ∈ 𝒫 ( O ‘𝐴))
10		eqidd 2737	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) = (𝐿 |s 𝑅))
11		breq1 5127	. . . . 5 ⊢ (𝑙 = 𝐿 → (𝑙 <<s 𝑟 ↔ 𝐿 <<s 𝑟))
12		oveq1 7417	. . . . . 6 ⊢ (𝑙 = 𝐿 → (𝑙 |s 𝑟) = (𝐿 |s 𝑟))
13	12	eqeq1d 2738	. . . . 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 5128	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐿 <<s 𝑟 ↔ 𝐿 <<s 𝑅))
16		oveq2 7418	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝐿 |s 𝑟) = (𝐿 |s 𝑅))
17	16	eqeq1d 2738	. . . . 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 3619	. . 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 27837	. . 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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  𝒫 cpw 4580   class class class wbr 5124  Oncon0 6357  ‘cfv 6536  (class class class)co 7410   <<s csslt 27749   |s cscut 27751   M cmade 27807   O cold 27808

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sslt 27750  df-scut 27752  df-made 27812  df-old 27813

This theorem is referenced by:  madebday  27868