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Theorem madeval2 27585
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 27584 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
2 scutcl 27540 . . . . . . . 8 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
3 eleq1 2819 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No 𝑥 No ))
43biimpd 228 . . . . . . . 8 ((𝑎 |s 𝑏) = 𝑥 → ((𝑎 |s 𝑏) ∈ No 𝑥 No ))
52, 4mpan9 505 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
65rexlimivw 3149 . . . . . 6 (∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
76rexlimivw 3149 . . . . 5 (∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
87pm4.71ri 559 . . . 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 2800 . . 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 2819 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11 breq1 5150 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
1210, 11anbi12d 629 . . . . . 6 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
1312rexxp 5841 . . . . 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 6297 . . . . . . . 8 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ))
15 dmscut 27549 . . . . . . . . . 10 dom |s = <<s
1615ineq2i 4208 . . . . . . . . 9 ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )
1716imaeq2i 6056 . . . . . . . 8 ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1814, 17eqtri 2758 . . . . . . 7 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1918eleq2i 2823 . . . . . 6 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )))
20 vex 3476 . . . . . . 7 𝑥 ∈ V
2120elima 6063 . . . . . 6 (𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22 elin 3963 . . . . . . . . 9 (𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ) ↔ (𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∧ 𝑦 ∈ <<s ))
2322anbi1i 622 . . . . . . . 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 467 . . . . . . . 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 274 . . . . . . 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 3088 . . . . . 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 296 . . . . 5 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ ∃𝑦 ∈ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))(𝑦 ∈ <<s ∧ 𝑦 |s 𝑥))
28 df-br 5148 . . . . . . . 8 (𝑎 <<s 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ <<s )
2928anbi1i 622 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥))
30 df-ov 7414 . . . . . . . . . 10 (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
3130eqeq1i 2735 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32 scutf 27550 . . . . . . . . . . 11 |s : <<s ⟶ No
33 ffn 6716 . . . . . . . . . . 11 ( |s : <<s ⟶ No → |s Fn <<s )
3432, 33ax-mp 5 . . . . . . . . . 10 |s Fn <<s
35 fnbrfvb 6943 . . . . . . . . . 10 (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3634, 35mpan 686 . . . . . . . . 9 (⟨𝑎, 𝑏⟩ ∈ <<s → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3731, 36bitrid 282 . . . . . . . 8 (⟨𝑎, 𝑏⟩ ∈ <<s → ((𝑎 |s 𝑏) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3837pm5.32i 573 . . . . . . 7 ((⟨𝑎, 𝑏⟩ ∈ <<s ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
3929, 38bitri 274 . . . . . 6 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥))
40392rexbii 3127 . . . . 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 302 . . . 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 2867 . . 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 3431 . . 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 2768 . 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 2786 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 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068  {crab 3430  cin 3946  𝒫 cpw 4601  cop 4633   cuni 4907   class class class wbr 5147   × cxp 5673  dom cdm 5675  cima 5678  Oncon0 6363   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7411   No csur 27379   <<s csslt 27518   |s cscut 27520   M cmade 27574
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-2o 8469  df-no 27382  df-slt 27383  df-bday 27384  df-sslt 27519  df-scut 27521  df-made 27579
This theorem is referenced by:  madef  27588  elmade  27599  made0  27605  madess  27608
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