Theorem madeval2 28252

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 28251	. 2 ⊢ (𝐴 ∈ On → (M‘𝐴) = ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))))
2		cutscl 28201	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No)
3		eleq1 2852	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No ↔ 𝑥 ∈ No))
4	3	biimpd 232	. . . . . . . 8 ⊢ ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No → 𝑥 ∈ No))
5	2, 4	mpan9 516	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
6	5	rexlimivw 3163	. . . . . 6 ⊢ (∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
7	6	rexlimivw 3163	. . . . 5 ⊢ (∃𝑎 ∈ 𝒫 ∪ (M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 ∈ No)
8	7	pm4.71ri 570	. . . 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 2830	. . 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 2852	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11		breq1 5109	. . . . . . 7 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
12	10, 11	anbi12d 644	. . . . . 6 ⊢ (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
13	12	rexxp 5826	. . . . 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 6323	. . . . . . . 8 ⊢ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s ))
15		dmcuts 28210	. . . . . . . . . 10 ⊢ dom |s = <<s
16	15	ineq2i 4117	. . . . . . . . 9 ⊢ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s ) = ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )
17	16	imaeq2i 6063	. . . . . . . 8 ⊢ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ))
18	14, 17	eqtri 2786	. . . . . . 7 ⊢ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) = ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ))
19	18	eleq2i 2856	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )))
20		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
21	20	elima 6074	. . . . . 6 ⊢ (𝑥 ∈ ( |s “ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22		elin 4096	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
23	22	anbi1i 636	. . . . . . . 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 474	. . . . . . . 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 278	. . . . . . 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 3109	. . . . . 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 300	. . . . 5 ⊢ (𝑥 ∈ ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
28		df-br 5107	. . . . . . . 8 ⊢ (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
29	28	anbi1i 636	. . . . . . 7 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
30		df-ov 7449	. . . . . . . . . 10 ⊢ (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
31	30	eqeq1i 2768	. . . . . . . . 9 ⊢ ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32		cutsf 28211	. . . . . . . . . . 11 ⊢ |s : <<s ⟶No
33		ffn 6734	. . . . . . . . . . 11 ⊢ ( |s : <<s ⟶No → |s Fn <<s )
34	32, 33	ax-mp 5	. . . . . . . . . 10 ⊢ |s Fn <<s
35		fnbrfvb 6961	. . . . . . . . . 10 ⊢ (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
36	34, 35	mpan 703	. . . . . . . . 9 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
37	31, 36	bitrid 286	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
38	37	pm5.32i 585	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
39	29, 38	bitri 278	. . . . . 6 ⊢ ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40	39	2rexbii 3142	. . . . 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 306	. . . 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 2899	. . 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 3417	. . 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 2796	. 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 2814	1 ⊢ (𝐴 ∈ On → (M‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ (M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ (M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  {crab 3416   ∩ cin 4092  𝒫 cpw 4560  ⟨cop 4838  ∪ cuni 4870   class class class wbr 5106   × cxp 5654  dom cdm 5656   “ cima 5659  Oncon0 6383   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <<s cslts 28176   |s ccuts 28178  Mcmade 28241

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246

This theorem is used by:  madef  28255  elmade  28276  made0  28282  madess  28285