Theorem elmade 28249

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 28228	. . . . 5 ⊢ M:On⟶𝒫 No
2	1	ffvelcdmi 7084	. . . 4 ⊢ (𝐴 ∈ On → (M‘𝐴) ∈ 𝒫 No)
3	2	elpwid 4565	. . 3 ⊢ (𝐴 ∈ On → (M‘𝐴) ⊆ No)
4	3	sseld 3907	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (M‘𝐴) → 𝑋 ∈ No))
5		cutscl 28174	. . . . . 6 ⊢ (𝑙 <<s 𝑟 → (𝑙 |s 𝑟) ∈ No)
6		eleq1 2848	. . . . . . 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 3159	. . . 4 ⊢ (∃𝑟 ∈ 𝒫 ∪ (M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋) → 𝑋 ∈ No)
10	9	rexlimivw 3159	. . 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 28225	. . . . 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 2846	. . . 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 2772	. . . . . . 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 3227	. . . . 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 3645	. . . 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 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102   “ cima 5653  Oncon0 6362  ‘cfv 6539  (class class class)co 7421  Nocsur 28003   <<s cslts 28149   |s ccuts 28151  Mcmade 28214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219

This theorem is used by:  elmade2  28250