Theorem madeval 32233

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 32228	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 7655	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)))
3	1	tfr1 7654	. . . . 5 ⊢ M Fn On
4		fnfun 6141	. . . . 5 ⊢ ( M Fn On → Fun M )
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun M
6		resfunexg 6635	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
7	5, 6	mpan 708	. . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
8		scutf 32217	. . . . 5 ⊢ |s : <<s ⟶ No
9		ffun 6201	. . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s )
10	8, 9	ax-mp 5	. . . 4 ⊢ Fun |s
11		funimaexg 6128	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
12	5, 11	mpan 708	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
13		uniexg 7112	. . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V)
14		pwexg 4991	. . . . . 6 ⊢ (∪ ( M “ 𝐴) ∈ V → 𝒫 ∪ ( M “ 𝐴) ∈ V)
15	12, 13, 14	3syl 18	. . . . 5 ⊢ (𝐴 ∈ On → 𝒫 ∪ ( M “ 𝐴) ∈ V)
16		xpexg 7117	. . . . 5 ⊢ ((𝒫 ∪ ( M “ 𝐴) ∈ V ∧ 𝒫 ∪ ( M “ 𝐴) ∈ V) → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
17	15, 15, 16	syl2anc 696	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
18		funimaexg 6128	. . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
19	10, 17, 18	sylancr 698	. . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
20		rneq 5498	. . . . . . . . 9 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
21		df-ima 5271	. . . . . . . . 9 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴)
22	20, 21	syl6eqr 2804	. . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
23	22	unieqd 4590	. . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴))
24	23	pweqd 4299	. . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴))
25	24	sqxpeqd 5290	. . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))
26	25	imaeq2d 5616	. . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
27		eqid 2752	. . . 4 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))
28	26, 27	fvmptg 6434	. . 3 ⊢ ((( M ↾ 𝐴) ∈ V ∧ ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
29	7, 19, 28	syl2anc 696	. 2 ⊢ (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
30	2, 29	eqtrd 2786	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))

Syntax hints:   → wi 4   = wceq 1624   ∈ wcel 2131  Vcvv 3332  𝒫 cpw 4294  ∪ cuni 4580   ↦ cmpt 4873   × cxp 5256  ran crn 5259   ↾ cres 5260   “ cima 5261  Oncon0 5876  Fun wfun 6035   Fn wfn 6036  ⟶wf 6037  ‘cfv 6041   No csur 32091   <<s csslt 32194   |s cscut 32196   M cmade 32223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-wrecs 7568  df-recs 7629  df-1o 7721  df-2o 7722  df-no 32094  df-slt 32095  df-bday 32096  df-sslt 32195  df-scut 32197  df-made 32228

This theorem is referenced by:  madeval2  32234