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Theorem madeval2 27907
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 27906 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
2 scutcl 27862 . . . . . . . 8 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
3 eleq1 2827 . . . . . . . . 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 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 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 2807 . . 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 2827 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11 breq1 5151 . . . . . . 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 5856 . . . . 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 6321 . . . . . . . 8 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ))
15 dmscut 27871 . . . . . . . . . 10 dom |s = <<s
1615ineq2i 4225 . . . . . . . . 9 ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )
1716imaeq2i 6078 . . . . . . . 8 ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1814, 17eqtri 2763 . . . . . . 7 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1918eleq2i 2831 . . . . . 6 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )))
20 vex 3482 . . . . . . 7 𝑥 ∈ V
2120elima 6085 . . . . . 6 (𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22 elin 3979 . . . . . . . . 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 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 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 5149 . . . . . . . 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 2740 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32 scutf 27872 . . . . . . . . . . 11 |s : <<s ⟶ No
33 ffn 6737 . . . . . . . . . . 11 ( |s : <<s ⟶ No → |s Fn <<s )
3432, 33ax-mp 5 . . . . . . . . . 10 |s Fn <<s
35 fnbrfvb 6960 . . . . . . . . . 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 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 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 2875 . . 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 3434 . . 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 2773 . 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 2791 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 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  cin 3962  𝒫 cpw 4605  cop 4637   cuni 4912   class class class wbr 5148   × cxp 5687  dom cdm 5689  cima 5692  Oncon0 6386   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431   No csur 27699   <<s csslt 27840   |s cscut 27842   M cmade 27896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901
This theorem is referenced by:  madef  27910  elmade  27921  made0  27927  madess  27930
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