Theorem madecut 33992

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 765	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 <<s 𝑅)
2		ssltex1 33908	. . . . 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 767	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ⊆ ( O ‘𝐴))
5	3, 4	elpwd 4538	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ 𝒫 ( O ‘𝐴))
6		ssltex2 33909	. . . . 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 769	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ⊆ ( O ‘𝐴))
9	7, 8	elpwd 4538	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝑅 ∈ 𝒫 ( O ‘𝐴))
10		eqidd 2739	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) = (𝐿 |s 𝑅))
11		breq1 5073	. . . . 5 ⊢ (𝑙 = 𝐿 → (𝑙 <<s 𝑟 ↔ 𝐿 <<s 𝑟))
12		oveq1 7262	. . . . . 6 ⊢ (𝑙 = 𝐿 → (𝑙 |s 𝑟) = (𝐿 |s 𝑟))
13	12	eqeq1d 2740	. . . . 5 ⊢ (𝑙 = 𝐿 → ((𝑙 |s 𝑟) = (𝐿 |s 𝑅) ↔ (𝐿 |s 𝑟) = (𝐿 |s 𝑅)))
14	11, 13	anbi12d 630	. . . 4 ⊢ (𝑙 = 𝐿 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅)) ↔ (𝐿 <<s 𝑟 ∧ (𝐿 |s 𝑟) = (𝐿 |s 𝑅))))
15		breq2 5074	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐿 <<s 𝑟 ↔ 𝐿 <<s 𝑅))
16		oveq2 7263	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝐿 |s 𝑟) = (𝐿 |s 𝑅))
17	16	eqeq1d 2740	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝐿 |s 𝑟) = (𝐿 |s 𝑅) ↔ (𝐿 |s 𝑅) = (𝐿 |s 𝑅)))
18	15, 17	anbi12d 630	. . . 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 3564	. . 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 1372	. 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 33979	. . 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 722	. 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 256	1 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  Oncon0 6251  ‘cfv 6418  (class class class)co 7255   <<s csslt 33902   |s cscut 33904   M cmade 33953   O cold 33954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959

This theorem is referenced by:  madebday  34007