Theorem madecut 27821

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 27719	. . . . 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 4554	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ 𝒫 ( O ‘𝐴))
6		ssltex2 27720	. . . . 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 4554	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ∈ 𝒫 ( O ‘𝐴))
10		eqidd 2731	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) = (𝐿 |s 𝑅))
11		breq1 5092	. . . . 5 ⊢ (𝑙 = 𝐿 → (𝑙 <<s 𝑟 ↔ 𝐿 <<s 𝑟))
12		oveq1 7348	. . . . . 6 ⊢ (𝑙 = 𝐿 → (𝑙 |s 𝑟) = (𝐿 |s 𝑟))
13	12	eqeq1d 2732	. . . . 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 5093	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐿 <<s 𝑟 ↔ 𝐿 <<s 𝑅))
16		oveq2 7349	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝐿 |s 𝑟) = (𝐿 |s 𝑅))
17	16	eqeq1d 2732	. . . . 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 3588	. . 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 27806	. . 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 1541   ∈ wcel 2110  ∃wrex 3054  Vcvv 3434   ⊆ wss 3900  𝒫 cpw 4548   class class class wbr 5089  Oncon0 6302  ‘cfv 6477  (class class class)co 7341   <<s csslt 27713   |s cscut 27715   M cmade 27776   O cold 27777

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 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-no 27574  df-slt 27575  df-bday 27576  df-sslt 27714  df-scut 27716  df-made 27781  df-old 27782

This theorem is referenced by:  madebday  27838