Theorem elmade 27813

Description: Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
elmade	⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))

Distinct variable groups:   𝐴,𝑙,𝑟   𝑋,𝑙,𝑟

Proof of Theorem elmade

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madef 27798	. . . . 5 ⊢ M :On⟶𝒫 No
2	1	ffvelcdmi 7016	. . . 4 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ∈ 𝒫 No )
3	2	elpwid 4559	. . 3 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ⊆ No )
4	3	sseld 3933	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → 𝑋 ∈ No ))
5		scutcl 27744	. . . . . 6 ⊢ (𝑙 <<s 𝑟 → (𝑙 |s 𝑟) ∈ No )
6		eleq1 2819	. . . . . . 7 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No ↔ 𝑋 ∈ No ))
7	6	biimpd 229	. . . . . 6 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No → 𝑋 ∈ No ))
8	5, 7	mpan9 506	. . . . 5 ⊢ ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
9	8	rexlimivw 3129	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
10	9	rexlimivw 3129	. . 3 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
11	10	a1i 11	. 2 ⊢ (𝐴 ∈ On → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No ))
12		madeval2 27795	. . . . 5 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
13	12	eleq2d 2817	. . . 4 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ 𝑋 ∈ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)}))
14		eqeq2 2743	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s 𝑟) = 𝑋))
15	14	anbi2d 630	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
16	15	2rexbidv 3197	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
17	16	elrab3 3648	. . . 4 ⊢ (𝑋 ∈ No → (𝑋 ∈ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
18	13, 17	sylan9bb 509	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
19	18	ex 412	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ No → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋))))
20	4, 11, 19	pm5.21ndd 379	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  𝒫 cpw 4550  ∪ cuni 4859   class class class wbr 5091   “ cima 5619  Oncon0 6306  ‘cfv 6481  (class class class)co 7346   No csur 27579   <<s csslt 27721   |s cscut 27723   M cmade 27784

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-no 27582  df-slt 27583  df-bday 27584  df-sslt 27722  df-scut 27724  df-made 27789

This theorem is referenced by:  elmade2  27814