Theorem madeval 28251

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 28246	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 8437	. 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 5926	. . . . . . . 8 ⊢ (𝑥 = (M ↾ 𝐴) → ran 𝑥 = ran (M ↾ 𝐴))
5		df-ima 5669	. . . . . . . 8 ⊢ (M “ 𝐴) = ran (M ↾ 𝐴)
6	4, 5	eqtr4di 2816	. . . . . . 7 ⊢ (𝑥 = (M ↾ 𝐴) → ran 𝑥 = (M “ 𝐴))
7	6	unieqd 4883	. . . . . 6 ⊢ (𝑥 = (M ↾ 𝐴) → ∪ ran 𝑥 = ∪ (M “ 𝐴))
8	7	pweqd 4577	. . . . 5 ⊢ (𝑥 = (M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ (M “ 𝐴))
9	8	sqxpeqd 5688	. . . 4 ⊢ (𝑥 = (M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)))
10	9	imaeq2d 6065	. . 3 ⊢ (𝑥 = (M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))))
11	1	tfr1 8436	. . . . 5 ⊢ M Fn On
12		fnfun 6664	. . . . 5 ⊢ (M Fn On → Fun M)
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun M
14		resfunexg 7247	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M ↾ 𝐴) ∈ V)
15	13, 14	mpan 703	. . 3 ⊢ (𝐴 ∈ On → (M ↾ 𝐴) ∈ V)
16		cutsf 28211	. . . . 5 ⊢ |s : <<s ⟶No
17		ffun 6737	. . . . 5 ⊢ ( |s : <<s ⟶No → Fun |s )
18	16, 17	ax-mp 5	. . . 4 ⊢ Fun |s
19		funimaexg 6651	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M “ 𝐴) ∈ V)
20	13, 19	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ On → (M “ 𝐴) ∈ V)
21		uniexg 7784	. . . . . 6 ⊢ ((M “ 𝐴) ∈ V → ∪ (M “ 𝐴) ∈ V)
22		pwexg 5345	. . . . . 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 7792	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∈ V)
25		funimaexg 6651	. . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ∈ V)
26	18, 24, 25	sylancr 599	. . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))) ∈ V)
27	3, 10, 15, 26	fvmptd3 7043	. 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))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5190   × cxp 5654  ran crn 5657   ↾ cres 5658   “ cima 5659  Oncon0 6383  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  Nocsur 28030   <<s cslts 28176   |s ccuts 28178  Mcmade 28241

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246

This theorem is used by:  madeval2  28252  madefi  28332