Theorem madeval 27337

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.

Step	Hyp	Ref	Expression
1		df-made 27332	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 8395	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)))
3		eqid 2733	. . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))
4		rneq 5934	. . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
5		df-ima 5689	. . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴)
6	4, 5	eqtr4di 2791	. . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
7	6	unieqd 4922	. . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴))
8	7	pweqd 4619	. . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴))
9	8	sqxpeqd 5708	. . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))
10	9	imaeq2d 6058	. . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
11	1	tfr1 8394	. . . . 5 ⊢ M Fn On
12		fnfun 6647	. . . . 5 ⊢ ( M Fn On → Fun M )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun M
14		resfunexg 7214	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
15	13, 14	mpan 689	. . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
16		scutf 27303	. . . . 5 ⊢ |s : <<s ⟶ No
17		ffun 6718	. . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s )
18	16, 17	ax-mp 5	. . . 4 ⊢ Fun |s
19		funimaexg 6632	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
20	13, 19	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
21		uniexg 7727	. . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V)
22		pwexg 5376	. . . . . 6 ⊢ (∪ ( M “ 𝐴) ∈ V → 𝒫 ∪ ( M “ 𝐴) ∈ V)
23	20, 21, 22	3syl 18	. . . . 5 ⊢ (𝐴 ∈ On → 𝒫 ∪ ( M “ 𝐴) ∈ V)
24	23, 23	xpexd 7735	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
25		funimaexg 6632	. . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
26	18, 24, 25	sylancr 588	. . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
27	3, 10, 15, 26	fvmptd3 7019	. 2 ⊢ (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
28	2, 27	eqtrd 2773	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475  𝒫 cpw 4602  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ran crn 5677   ↾ cres 5678   “ cima 5679  Oncon0 6362  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541   No csur 27133   <<s csslt 27272   |s cscut 27274   M cmade 27327

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 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sslt 27273  df-scut 27275  df-made 27332

This theorem is referenced by:  madeval2  27338