Theorem elmade 27823

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 27812	. . . . 5 ⊢ M :On⟶𝒫 No
2	1	ffvelcdmi 7072	. . . 4 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ∈ 𝒫 No )
3	2	elpwid 4584	. . 3 ⊢ (𝐴 ∈ On → ( M ‘𝐴) ⊆ No )
4	3	sseld 3957	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → 𝑋 ∈ No ))
5		scutcl 27764	. . . . . 6 ⊢ (𝑙 <<s 𝑟 → (𝑙 |s 𝑟) ∈ No )
6		eleq1 2822	. . . . . . 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 3137	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No )
10	9	rexlimivw 3137	. . 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 27809	. . . . 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 2820	. . . 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 2747	. . . . . . 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 3206	. . . . 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 3672	. . . 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 1540   ∈ wcel 2108  ∃wrex 3060  {crab 3415  𝒫 cpw 4575  ∪ cuni 4883   class class class wbr 5119   “ cima 5657  Oncon0 6352  ‘cfv 6530  (class class class)co 7403   No csur 27601   <<s csslt 27742   |s cscut 27744   M cmade 27798

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-no 27604  df-slt 27605  df-bday 27606  df-sslt 27743  df-scut 27745  df-made 27803

This theorem is referenced by:  elmade2  27824