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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   <
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