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Theorem madeval 32400
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 32395 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr2 7702 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)))
31tfr1 7701 . . . . 5 M Fn On
4 fnfun 6168 . . . . 5 ( M Fn On → Fun M )
53, 4ax-mp 5 . . . 4 Fun M
6 resfunexg 6676 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
75, 6mpan 681 . . 3 (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
8 scutf 32384 . . . . 5 |s : <<s ⟶ No
9 ffun 6228 . . . . 5 ( |s : <<s ⟶ No → Fun |s )
108, 9ax-mp 5 . . . 4 Fun |s
11 funimaexg 6155 . . . . . . 7 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
125, 11mpan 681 . . . . . 6 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
13 uniexg 7157 . . . . . 6 (( M “ 𝐴) ∈ V → ( M “ 𝐴) ∈ V)
14 pwexg 5016 . . . . . 6 ( ( M “ 𝐴) ∈ V → 𝒫 ( M “ 𝐴) ∈ V)
1512, 13, 143syl 18 . . . . 5 (𝐴 ∈ On → 𝒫 ( M “ 𝐴) ∈ V)
16 xpexg 7162 . . . . 5 ((𝒫 ( M “ 𝐴) ∈ V ∧ 𝒫 ( M “ 𝐴) ∈ V) → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
1715, 15, 16syl2anc 579 . . . 4 (𝐴 ∈ On → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
18 funimaexg 6155 . . . 4 ((Fun |s ∧ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
1910, 17, 18sylancr 581 . . 3 (𝐴 ∈ On → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
20 rneq 5521 . . . . . . . . 9 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
21 df-ima 5292 . . . . . . . . 9 ( M “ 𝐴) = ran ( M ↾ 𝐴)
2220, 21syl6eqr 2817 . . . . . . . 8 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2322unieqd 4606 . . . . . . 7 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2423pweqd 4322 . . . . . 6 (𝑥 = ( M ↾ 𝐴) → 𝒫 ran 𝑥 = 𝒫 ( M “ 𝐴))
2524sqxpeqd 5311 . . . . 5 (𝑥 = ( M ↾ 𝐴) → (𝒫 ran 𝑥 × 𝒫 ran 𝑥) = (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)))
2625imaeq2d 5650 . . . 4 (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
27 eqid 2765 . . . 4 (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))
2826, 27fvmptg 6473 . . 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 579 . 2 (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
302, 29eqtrd 2799 1 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  𝒫 cpw 4317   cuni 4596  cmpt 4890   × cxp 5277  ran crn 5280  cres 5281  cima 5282  Oncon0 5910  Fun wfun 6064   Fn wfn 6065  wf 6066  cfv 6070   No csur 32258   <<s csslt 32361   |s cscut 32363   M cmade 32390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-wrecs 7614  df-recs 7676  df-1o 7768  df-2o 7769  df-no 32261  df-slt 32262  df-bday 32263  df-sslt 32362  df-scut 32364  df-made 32395
This theorem is referenced by:  madeval2  32401
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