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Theorem madeval2 27183
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 27182 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
2 scutcl 27141 . . . . . . . 8 (𝑎 <<s 𝑏 → (𝑎 |s 𝑏) ∈ No )
3 eleq1 2825 . . . . . . . . 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 507 . . . . . . 7 ((𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
65rexlimivw 3148 . . . . . 6 (∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
76rexlimivw 3148 . . . . 5 (∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥) → 𝑥 No )
87pm4.71ri 561 . . . 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 2806 . . 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 2825 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 ∈ <<s ↔ ⟨𝑎, 𝑏⟩ ∈ <<s ))
11 breq1 5108 . . . . . . 7 (𝑦 = ⟨𝑎, 𝑏⟩ → (𝑦 |s 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
1210, 11anbi12d 631 . . . . . 6 (𝑦 = ⟨𝑎, 𝑏⟩ → ((𝑦 ∈ <<s ∧ 𝑦 |s 𝑥) ↔ (⟨𝑎, 𝑏⟩ ∈ <<s ∧ ⟨𝑎, 𝑏⟩ |s 𝑥)))
1312rexxp 5798 . . . . 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 6251 . . . . . . . 8 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ))
15 dmscut 27150 . . . . . . . . . 10 dom |s = <<s
1615ineq2i 4169 . . . . . . . . 9 ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s ) = ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )
1716imaeq2i 6011 . . . . . . . 8 ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ dom |s )) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1814, 17eqtri 2764 . . . . . . 7 ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) = ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s ))
1918eleq2i 2829 . . . . . 6 (𝑥 ∈ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ↔ 𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )))
20 vex 3449 . . . . . . 7 𝑥 ∈ V
2120elima 6018 . . . . . 6 (𝑥 ∈ ( |s “ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )) ↔ ∃𝑦 ∈ ((𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∩ <<s )𝑦 |s 𝑥)
22 elin 3926 . . . . . . . . 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 469 . . . . . . . 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 3093 . . . . . 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 5106 . . . . . . . 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 7360 . . . . . . . . . 10 (𝑎 |s 𝑏) = ( |s ‘⟨𝑎, 𝑏⟩)
3130eqeq1i 2741 . . . . . . . . 9 ((𝑎 |s 𝑏) = 𝑥 ↔ ( |s ‘⟨𝑎, 𝑏⟩) = 𝑥)
32 scutf 27151 . . . . . . . . . . 11 |s : <<s ⟶ No
33 ffn 6668 . . . . . . . . . . 11 ( |s : <<s ⟶ No → |s Fn <<s )
3432, 33ax-mp 5 . . . . . . . . . 10 |s Fn <<s
35 fnbrfvb 6895 . . . . . . . . . 10 (( |s Fn <<s ∧ ⟨𝑎, 𝑏⟩ ∈ <<s ) → (( |s ‘⟨𝑎, 𝑏⟩) = 𝑥 ↔ ⟨𝑎, 𝑏⟩ |s 𝑥))
3634, 35mpan 688 . . . . . . . . 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 575 . . . . . . 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 3128 . . . . 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))
4241abbi2i 2873 . . 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 3408 . . 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 2774 . 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 2792 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 396   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  {crab 3407  cin 3909  𝒫 cpw 4560  cop 4592   cuni 4865   class class class wbr 5105   × cxp 5631  dom cdm 5633  cima 5636  Oncon0 6317   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357   No csur 26988   <<s csslt 27120   |s cscut 27122   M cmade 27172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sslt 27121  df-scut 27123  df-made 27177
This theorem is referenced by:  madef  27186  elmade  27197  made0  27203  madess  27206
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