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Theorem madeval 32233
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 32228 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr2 7655 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)))
31tfr1 7654 . . . . 5 M Fn On
4 fnfun 6141 . . . . 5 ( M Fn On → Fun M )
53, 4ax-mp 5 . . . 4 Fun M
6 resfunexg 6635 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
75, 6mpan 708 . . 3 (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
8 scutf 32217 . . . . 5 |s : <<s ⟶ No
9 ffun 6201 . . . . 5 ( |s : <<s ⟶ No → Fun |s )
108, 9ax-mp 5 . . . 4 Fun |s
11 funimaexg 6128 . . . . . . 7 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
125, 11mpan 708 . . . . . 6 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
13 uniexg 7112 . . . . . 6 (( M “ 𝐴) ∈ V → ( M “ 𝐴) ∈ V)
14 pwexg 4991 . . . . . 6 ( ( M “ 𝐴) ∈ V → 𝒫 ( M “ 𝐴) ∈ V)
1512, 13, 143syl 18 . . . . 5 (𝐴 ∈ On → 𝒫 ( M “ 𝐴) ∈ V)
16 xpexg 7117 . . . . 5 ((𝒫 ( M “ 𝐴) ∈ V ∧ 𝒫 ( M “ 𝐴) ∈ V) → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
1715, 15, 16syl2anc 696 . . . 4 (𝐴 ∈ On → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
18 funimaexg 6128 . . . 4 ((Fun |s ∧ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
1910, 17, 18sylancr 698 . . 3 (𝐴 ∈ On → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
20 rneq 5498 . . . . . . . . 9 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
21 df-ima 5271 . . . . . . . . 9 ( M “ 𝐴) = ran ( M ↾ 𝐴)
2220, 21syl6eqr 2804 . . . . . . . 8 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2322unieqd 4590 . . . . . . 7 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
2423pweqd 4299 . . . . . 6 (𝑥 = ( M ↾ 𝐴) → 𝒫 ran 𝑥 = 𝒫 ( M “ 𝐴))
2524sqxpeqd 5290 . . . . 5 (𝑥 = ( M ↾ 𝐴) → (𝒫 ran 𝑥 × 𝒫 ran 𝑥) = (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)))
2625imaeq2d 5616 . . . 4 (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
27 eqid 2752 . . . 4 (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))
2826, 27fvmptg 6434 . . 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 696 . 2 (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
302, 29eqtrd 2786 1 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  Vcvv 3332  𝒫 cpw 4294   cuni 4580  cmpt 4873   × cxp 5256  ran crn 5259  cres 5260  cima 5261  Oncon0 5876  Fun wfun 6035   Fn wfn 6036  wf 6037  cfv 6041   No csur 32091   <<s csslt 32194   |s cscut 32196   M cmade 32223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-wrecs 7568  df-recs 7629  df-1o 7721  df-2o 7722  df-no 32094  df-slt 32095  df-bday 32096  df-sslt 32195  df-scut 32197  df-made 32228
This theorem is referenced by:  madeval2  32234
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