Theorem madecut 27865

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 769	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 <<s 𝑅)
2		ssltex1 27763	. . . . 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 4561	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → 𝐿 ∈ 𝒫 ( O ‘𝐴))
6		ssltex2 27764	. . . . 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 4561	. . 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 5102	. . . . 5 ⊢ (𝑙 = 𝐿 → (𝑙 <<s 𝑟 ↔ 𝐿 <<s 𝑟))
12		oveq1 7367	. . . . . 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 633	. . . 4 ⊢ (𝑙 = 𝐿 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = (𝐿 |s 𝑅)) ↔ (𝐿 <<s 𝑟 ∧ (𝐿 |s 𝑟) = (𝐿 |s 𝑅))))
15		breq2 5103	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐿 <<s 𝑟 ↔ 𝐿 <<s 𝑅))
16		oveq2 7368	. . . . . 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 633	. . . 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 3590	. . 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 1377	. 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 27850	. . 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 727	. 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 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441   ⊆ wss 3902  𝒫 cpw 4555   class class class wbr 5099  Oncon0 6318  ‘cfv 6493  (class class class)co 7360   <<s csslt 27757   |s cscut 27759   M cmade 27820   O cold 27821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27614  df-slt 27615  df-bday 27616  df-sslt 27758  df-scut 27760  df-made 27825  df-old 27826

This theorem is referenced by:  madebday  27882