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Theorem madeval 27185
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 27180 . . 3 M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))))
21tfr2 8345 . 2 (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)))
3 eqid 2737 . . 3 (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))
4 rneq 5892 . . . . . . . 8 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
5 df-ima 5647 . . . . . . . 8 ( M “ 𝐴) = ran ( M ↾ 𝐴)
64, 5eqtr4di 2795 . . . . . . 7 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
76unieqd 4880 . . . . . 6 (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
87pweqd 4578 . . . . 5 (𝑥 = ( M ↾ 𝐴) → 𝒫 ran 𝑥 = 𝒫 ( M “ 𝐴))
98sqxpeqd 5666 . . . 4 (𝑥 = ( M ↾ 𝐴) → (𝒫 ran 𝑥 × 𝒫 ran 𝑥) = (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)))
109imaeq2d 6014 . . 3 (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
111tfr1 8344 . . . . 5 M Fn On
12 fnfun 6603 . . . . 5 ( M Fn On → Fun M )
1311, 12ax-mp 5 . . . 4 Fun M
14 resfunexg 7166 . . . 4 ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
1513, 14mpan 689 . . 3 (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
16 scutf 27154 . . . . 5 |s : <<s ⟶ No
17 ffun 6672 . . . . 5 ( |s : <<s ⟶ No → Fun |s )
1816, 17ax-mp 5 . . . 4 Fun |s
19 funimaexg 6588 . . . . . . 7 ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
2013, 19mpan 689 . . . . . 6 (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
21 uniexg 7678 . . . . . 6 (( M “ 𝐴) ∈ V → ( M “ 𝐴) ∈ V)
22 pwexg 5334 . . . . . 6 ( ( M “ 𝐴) ∈ V → 𝒫 ( M “ 𝐴) ∈ V)
2320, 21, 223syl 18 . . . . 5 (𝐴 ∈ On → 𝒫 ( M “ 𝐴) ∈ V)
2423, 23xpexd 7686 . . . 4 (𝐴 ∈ On → (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V)
25 funimaexg 6588 . . . 4 ((Fun |s ∧ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
2618, 24, 25sylancr 588 . . 3 (𝐴 ∈ On → ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))) ∈ V)
273, 10, 15, 26fvmptd3 6972 . 2 (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ran 𝑥 × 𝒫 ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
282, 27eqtrd 2777 1 (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3446  𝒫 cpw 4561   cuni 4866  cmpt 5189   × cxp 5632  ran crn 5635  cres 5636  cima 5637  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  wf 6493  cfv 6497   No csur 26991   <<s csslt 27123   |s cscut 27125   M cmade 27175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-made 27180
This theorem is referenced by:  madeval2  27186
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