Theorem madeval 28006

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 28001	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 8386	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)))
3		eqid 2763	. . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))
4		rneq 5928	. . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
5		df-ima 5676	. . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴)
6	4, 5	eqtr4di 2816	. . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
7	6	unieqd 4886	. . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴))
8	7	pweqd 4580	. . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴))
9	8	sqxpeqd 5695	. . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))
10	9	imaeq2d 6064	. . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
11	1	tfr1 8385	. . . . 5 ⊢ M Fn On
12		fnfun 6637	. . . . 5 ⊢ ( M Fn On → Fun M )
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun M
14		resfunexg 7215	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
15	13, 14	mpan 702	. . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
16		cutsf 27966	. . . . 5 ⊢ |s : <<s ⟶ No
17		ffun 6710	. . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s )
18	16, 17	ax-mp 5	. . . 4 ⊢ Fun |s
19		funimaexg 6624	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
20	13, 19	mpan 702	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
21		uniexg 7740	. . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V)
22		pwexg 5351	. . . . . 6 ⊢ (∪ ( M “ 𝐴) ∈ V → 𝒫 ∪ ( M “ 𝐴) ∈ V)
23	20, 21, 22	3syl 19	. . . . 5 ⊢ (𝐴 ∈ On → 𝒫 ∪ ( M “ 𝐴) ∈ V)
24	23, 23	xpexd 7751	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
25		funimaexg 6624	. . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
26	18, 24, 25	sylancr 598	. . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
27	3, 10, 15, 26	fvmptd3 7015	. 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 2798	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  𝒫 cpw 4563  ∪ cuni 4873   ↦ cmpt 5193   × cxp 5661  ran crn 5664   ↾ cres 5665   “ cima 5666  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538   No csur 27785   <<s cslts 27931   |s ccuts 27933   M cmade 27996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-no 27788  df-lts 27789  df-bday 27790  df-slts 27932  df-cuts 27934  df-made 28001

This theorem is referenced by:  madeval2  28007  madefi  28087