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Theorem madeval2 32061
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.
StepHypRef Expression
1 madeval 32060 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
2 scutcut 32037 . . . . . . . . 9 (𝑎 <<s 𝑏 → ((𝑎 |s 𝑏) ∈ No 𝑎 <<s {(𝑎 |s 𝑏)} ∧ {(𝑎 |s 𝑏)} <<s 𝑏))
32simp1d 1093 . . . . . . . 8 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
4 eleq1 2718 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No 𝑥 No ))
54biimpd 219 . . . . . . . 8 ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No 𝑥 No ))
63, 5mpan9 485 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
76rexlimivw 3058 . . . . . 6 (∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
87rexlimivw 3058 . . . . 5 (∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
98pm4.71ri 666 . . . 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)))
109abbii 2768 . . 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))}
11 eleq1 2718 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
12 breq1 4688 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
1311, 12anbi12d 747 . . . . . 6 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
1413rexxp 5297 . . . . 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))
15 imaindm 31806 . . . . . . . 8 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ))
16 dmscut 32043 . . . . . . . . . 10 dom |s = <<s
1716ineq2i 3844 . . . . . . . . 9 ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )
1817imaeq2i 5499 . . . . . . . 8 ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1915, 18eqtri 2673 . . . . . . 7 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
2019eleq2i 2722 . . . . . 6 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )))
21 vex 3234 . . . . . . 7 𝑥 ∈ V
2221elima 5506 . . . . . 6 (𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
23 elin 3829 . . . . . . . . 9 (𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
2423anbi1i 731 . . . . . . . 8 ((𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ ((𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ) ∧ 𝑦 |s 𝑥))
25 anass 682 . . . . . . . 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)))
2624, 25bitri 264 . . . . . . 7 ((𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ) ∧ 𝑦 |s 𝑥) ↔ (𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∧ (𝑦 ∈ <<s ∧ 𝑦 |s 𝑥)))
2726rexbii2 3068 . . . . . 6 (∃𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥 ↔ ∃𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
2820, 22, 273bitri 286 . . . . 5 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
29 df-br 4686 . . . . . . . 8 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
3029anbi1i 731 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
31 df-ov 6693 . . . . . . . . . 10 (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
3231eqeq1i 2656 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
33 scutf 32044 . . . . . . . . . . 11 |s : <<s ⟶ No
34 ffn 6083 . . . . . . . . . . 11 ( |s : <<s ⟶ No → |s Fn <<s )
3533, 34ax-mp 5 . . . . . . . . . 10 |s Fn <<s
36 fnbrfvb 6274 . . . . . . . . . 10 (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3735, 36mpan 706 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3832, 37syl5bb 272 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3938pm5.32i 670 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
4030, 39bitri 264 . . . . . 6 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
41402rexbii 3071 . . . . 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))
4214, 28, 413bitr4i 292 . . . 4 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ ∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥))
4342abbi2i 2767 . . 3 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = {𝑥 ∣ ∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}
44 df-rab 2950 . . 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))}
4510, 43, 443eqtr4i 2683 . 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)}
461, 45syl6eq 2701 1 (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 No ∣ ∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  {crab 2945  cin 3606  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685   × cxp 5141  dom cdm 5143  cima 5146  Oncon0 5761   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690   No csur 31918   <<s csslt 32021   |s cscut 32023   M cmade 32050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-bday 31923  df-sslt 32022  df-scut 32024  df-made 32055
This theorem is referenced by: (None)
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