Theorem elmade 28300

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 28279	. . . . 5 ⊢ M:On⟶𝒫 No
2	1	ffvelcdmi 7115	. . . 4 ⊢ (𝐴 ∈ On → (M‘𝐴) ∈ 𝒫 No)
3	2	elpwid 4569	. . 3 ⊢ (𝐴 ∈ On → (M‘𝐴) ⊆ No)
4	3	sseld 3912	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (M‘𝐴) → 𝑋 ∈ No))
5		cutscl 28225	. . . . . 6 ⊢ (𝑙 <<s 𝑟 → (𝑙 |s 𝑟) ∈ No)
6		eleq1 2852	. . . . . . 7 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No ↔ 𝑋 ∈ No))
7	6	biimpd 232	. . . . . 6 ⊢ ((𝑙 |s 𝑟) = 𝑋 → ((𝑙 |s 𝑟) ∈ No → 𝑋 ∈ No))
8	5, 7	mpan9 516	. . . . 5 ⊢ ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No)
9	8	rexlimivw 3163	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 ∪ (M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No)
10	9	rexlimivw 3163	. . 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 28276	. . . . 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 2850	. . . 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 2775	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s 𝑟) = 𝑋))
15	14	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
16	15	2rexbidv 3231	. . . . 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 3649	. . . 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 519	. . 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 418	. 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 382	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ (M‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ (M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ (M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108   “ cima 5661  Oncon0 6388  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   <<s cslts 28200   |s ccuts 28202  Mcmade 28265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270

This theorem is used by:  elmade2  28301