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Theorem madeval 32265
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 32260 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr2 7645 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)))
31tfr1 7644 . . . . 5 M Fn On
4 fnfun 6126 . . . . 5 ( M Fn On → Fun M )
53, 4ax-mp 5 . . . 4 Fun M
6 resfunexg 6621 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
75, 6mpan 670 . . 3 (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
8 scutf 32249 . . . . 5 |s : <<s ⟶ No
9 ffun 6186 . . . . 5 ( |s : <<s ⟶ No → Fun |s )
108, 9ax-mp 5 . . . 4 Fun |s
11 funimaexg 6113 . . . . . . 7 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
125, 11mpan 670 . . . . . 6 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
13 uniexg 7100 . . . . . 6 (( M “ 𝐴) ∈ V → ( M “ 𝐴) ∈ V)
14 pwexg 4980 . . . . . 6 ( ( M “ 𝐴) ∈ V → 𝒫 ( M “ 𝐴) ∈ V)
1512, 13, 143syl 18 . . . . 5 (𝐴 ∈ On → 𝒫 ( M “ 𝐴) ∈ V)
16 xpexg 7105 . . . . 5 ((𝒫 ( M “ 𝐴) ∈ V ∧ 𝒫 ( M “ 𝐴) ∈ V) → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
1715, 15, 16syl2anc 573 . . . 4 (𝐴 ∈ On → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
18 funimaexg 6113 . . . 4 ((Fun |s ∧ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
1910, 17, 18sylancr 575 . . 3 (𝐴 ∈ On → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
20 rneq 5487 . . . . . . . . 9 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
21 df-ima 5262 . . . . . . . . 9 ( M “ 𝐴) = ran ( M ↾ 𝐴)
2220, 21syl6eqr 2823 . . . . . . . 8 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2322unieqd 4584 . . . . . . 7 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2423pweqd 4302 . . . . . 6 (𝑥 = ( M ↾ 𝐴) → 𝒫 ran 𝑥 = 𝒫 ( M “ 𝐴))
2524sqxpeqd 5280 . . . . 5 (𝑥 = ( M ↾ 𝐴) → (𝒫 ran 𝑥 × 𝒫 ran 𝑥) = (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)))
2625imaeq2d 5605 . . . 4 (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
27 eqid 2771 . . . 4 (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))
2826, 27fvmptg 6420 . . 3 ((( M ↾ 𝐴) ∈ V ∧ ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V) → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
297, 19, 28syl2anc 573 . 2 (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
302, 29eqtrd 2805 1 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  𝒫 cpw 4297   cuni 4574  cmpt 4863   × cxp 5247  ran crn 5250  cres 5251  cima 5252  Oncon0 5864  Fun wfun 6023   Fn wfn 6024  wf 6025  cfv 6029   No csur 32123   <<s csslt 32226   |s cscut 32228   M cmade 32255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-wrecs 7557  df-recs 7619  df-1o 7711  df-2o 7712  df-no 32126  df-slt 32127  df-bday 32128  df-sslt 32227  df-scut 32229  df-made 32260
This theorem is referenced by:  madeval2  32266
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