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Theorem madeval 33192
Description: The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
Assertion
Ref Expression
madeval (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))

Proof of Theorem madeval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-made 33187 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr2 8030 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)))
3 eqid 2826 . . 3 (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))
4 rneq 5805 . . . . . . . 8 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
5 df-ima 5567 . . . . . . . 8 ( M “ 𝐴) = ran ( M ↾ 𝐴)
64, 5syl6eqr 2879 . . . . . . 7 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
76unieqd 4847 . . . . . 6 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
87pweqd 4547 . . . . 5 (𝑥 = ( M ↾ 𝐴) → 𝒫 ran 𝑥 = 𝒫 ( M “ 𝐴))
98sqxpeqd 5586 . . . 4 (𝑥 = ( M ↾ 𝐴) → (𝒫 ran 𝑥 × 𝒫 ran 𝑥) = (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)))
109imaeq2d 5928 . . 3 (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
111tfr1 8029 . . . . 5 M Fn On
12 fnfun 6452 . . . . 5 ( M Fn On → Fun M )
1311, 12ax-mp 5 . . . 4 Fun M
14 resfunexg 6975 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
1513, 14mpan 686 . . 3 (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
16 scutf 33176 . . . . 5 |s : <<s ⟶ No
17 ffun 6516 . . . . 5 ( |s : <<s ⟶ No → Fun |s )
1816, 17ax-mp 5 . . . 4 Fun |s
19 funimaexg 6439 . . . . . . 7 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
2013, 19mpan 686 . . . . . 6 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
21 uniexg 7461 . . . . . 6 (( M “ 𝐴) ∈ V → ( M “ 𝐴) ∈ V)
22 pwexg 5276 . . . . . 6 ( ( M “ 𝐴) ∈ V → 𝒫 ( M “ 𝐴) ∈ V)
2320, 21, 223syl 18 . . . . 5 (𝐴 ∈ On → 𝒫 ( M “ 𝐴) ∈ V)
2423, 23xpexd 7467 . . . 4 (𝐴 ∈ On → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
25 funimaexg 6439 . . . 4 ((Fun |s ∧ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
2618, 24, 25sylancr 587 . . 3 (𝐴 ∈ On → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
273, 10, 15, 26fvmptd3 6789 . 2 (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
282, 27eqtrd 2861 1 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3500  𝒫 cpw 4542   cuni 4837  cmpt 5143   × cxp 5552  ran crn 5555  cres 5556  cima 5557  Oncon0 6190  Fun wfun 6348   Fn wfn 6349  wf 6350  cfv 6354   No csur 33050   <<s csslt 33153   |s cscut 33155   M cmade 33182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7943  df-recs 8004  df-1o 8098  df-2o 8099  df-no 33053  df-slt 33054  df-bday 33055  df-sslt 33154  df-scut 33156  df-made 33187
This theorem is referenced by:  madeval2  33193
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