Theorem madeval2 27892

Description: Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
madeval2	⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

Distinct variable group:   𝑥,𝐴,𝑎,𝑏

Proof of Theorem madeval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		madeval 27891	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
2		scutcl 27847	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
3		eleq1 2829	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No ↔ 𝑥 ∈ No ))
4	3	biimpd 229	. . . . . . . 8 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No → 𝑥 ∈ No ))
5	2, 4	mpan9 506	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
6	5	rexlimivw 3151	. . . . . 6 ⊢ (∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
7	6	rexlimivw 3151	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No )
8	7	pm4.71ri 560	. . . 4 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)))
9	8	abbii 2809	. . 3 ⊢ {𝑥 ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))}
10		eleq1 2829	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11		breq1 5146	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
12	10, 11	anbi12d 632	. . . . . 6 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
13	12	rexxp 5853	. . . . 5 ⊢ (∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
14		imaindm 6319	. . . . . . . 8 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s ))
15		dmscut 27856	. . . . . . . . . 10 ⊢ dom |s = <<s
16	15	ineq2i 4217	. . . . . . . . 9 ⊢ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )
17	16	imaeq2i 6076	. . . . . . . 8 ⊢ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ))
18	14, 17	eqtri 2765	. . . . . . 7 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ))
19	18	eleq2i 2833	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )))
20		vex 3484	. . . . . . 7 ⊢ 𝑥 ∈ V
21	20	elima 6083	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22		elin 3967	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
23	22	anbi1i 624	. . . . . . . 8 ⊢ ((𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ ((𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥))
24		anass 468	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ (𝑦 ∈ <<s ∧ 𝑦 |s 𝑥)))
25	23, 24	bitri 275	. . . . . . 7 ⊢ ((𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ (𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∧ (𝑦 ∈ <<s ∧ 𝑦 |s 𝑥)))
26	25	rexbii2 3090	. . . . . 6 ⊢ (∃𝑦 ∈ ((𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥 ↔ ∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
27	19, 21, 26	3bitri 297	. . . . 5 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
28		df-br 5144	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
29	28	anbi1i 624	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
30		df-ov 7434	. . . . . . . . . 10 ⊢ (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
31	30	eqeq1i 2742	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32		scutf 27857	. . . . . . . . . . 11 ⊢ |s : <<s ⟶ No
33		ffn 6736	. . . . . . . . . . 11 ⊢ ( |s : <<s ⟶ No → |s Fn <<s )
34	32, 33	ax-mp 5	. . . . . . . . . 10 ⊢ |s Fn <<s
35		fnbrfvb 6959	. . . . . . . . . 10 ⊢ (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
36	34, 35	mpan 690	. . . . . . . . 9 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
37	31, 36	bitrid 283	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
38	37	pm5.32i 574	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
39	29, 38	bitri 275	. . . . . 6 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40	39	2rexbii 3129	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
41	13, 27, 40	3bitr4i 303	. . . 4 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ↔ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))
42	41	eqabi 2877	. . 3 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = {𝑥 ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}
43		df-rab 3437	. . 3 ⊢ {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)} = {𝑥 ∣ (𝑥 ∈ No ∧ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))}
44	9, 42, 43	3eqtr4i 2775	. 2 ⊢ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}
45	1, 44	eqtrdi 2793	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  {crab 3436   ∩ cin 3950  𝒫 cpw 4600  ⟨cop 4632  ∪ cuni 4907   class class class wbr 5143   × cxp 5683  dom cdm 5685   “ cima 5688  Oncon0 6384   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   No csur 27684   <<s csslt 27825   |s cscut 27827   M cmade 27881

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886

This theorem is referenced by:  madef  27895  elmade  27906  made0  27912  madess  27915