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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.
StepHypRef 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 ))
43biimpd 229 . . . . . . . 8 ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No 𝑥 No ))
52, 4mpan9 506 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
65rexlimivw 3151 . . . . . 6 (∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
76rexlimivw 3151 . . . . 5 (∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
87pm4.71ri 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)))
98abbii 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))
1210, 11anbi12d 632 . . . . . 6 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
1312rexxp 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
1615ineq2i 4217 . . . . . . . . 9 ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )
1716imaeq2i 6076 . . . . . . . 8 ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1814, 17eqtri 2765 . . . . . . 7 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1918eleq2i 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
2120elima 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 ))
2322anbi1i 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)))
2523, 24bitri 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)))
2625rexbii2 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))
2719, 21, 263bitri 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 )
2928anbi1i 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⟩)
3130eqeq1i 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 )
3432, 33ax-mp 5 . . . . . . . . . 10 |s Fn <<s
35 fnbrfvb 6959 . . . . . . . . . 10 (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3634, 35mpan 690 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3731, 36bitrid 283 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3837pm5.32i 574 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
3929, 38bitri 275 . . . . . 6 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40392rexbii 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))
4113, 27, 403bitr4i 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))
4241eqabi 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))}
449, 42, 433eqtr4i 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)}
451, 44eqtrdi 2793 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 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
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