Theorem madeval 32265

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 32260	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 7645	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)))
3	1	tfr1 7644	. . . . 5 ⊢ M Fn On
4		fnfun 6126	. . . . 5 ⊢ ( M Fn On → Fun M )
5	3, 4	ax-mp 5	. . . 4 ⊢ Fun M
6		resfunexg 6621	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V)
7	5, 6	mpan 670	. . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V)
8		scutf 32249	. . . . 5 ⊢ |s : <<s ⟶ No
9		ffun 6186	. . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s )
10	8, 9	ax-mp 5	. . . 4 ⊢ Fun |s
11		funimaexg 6113	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
12	5, 11	mpan 670	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
13		uniexg 7100	. . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V)
14		pwexg 4980	. . . . . 6 ⊢ (∪ ( M “ 𝐴) ∈ V → 𝒫 ∪ ( M “ 𝐴) ∈ V)
15	12, 13, 14	3syl 18	. . . . 5 ⊢ (𝐴 ∈ On → 𝒫 ∪ ( M “ 𝐴) ∈ V)
16		xpexg 7105	. . . . 5 ⊢ ((𝒫 ∪ ( M “ 𝐴) ∈ V ∧ 𝒫 ∪ ( M “ 𝐴) ∈ V) → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
17	15, 15, 16	syl2anc 573	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V)
18		funimaexg 6113	. . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
19	10, 17, 18	sylancr 575	. . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V)
20		rneq 5487	. . . . . . . . 9 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴))
21		df-ima 5262	. . . . . . . . 9 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴)
22	20, 21	syl6eqr 2823	. . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴))
23	22	unieqd 4584	. . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴))
24	23	pweqd 4302	. . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴))
25	24	sqxpeqd 5280	. . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))
26	25	imaeq2d 5605	. . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))
27		eqid 2771	. . . 4 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))
28	26, 27	fvmptg 6420	. . 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 573	. 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 2805	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))))

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  Vcvv 3351  𝒫 cpw 4297  ∪ cuni 4574   ↦ cmpt 4863   × cxp 5247  ran crn 5250   ↾ cres 5251   “ cima 5252  Oncon0 5864  Fun wfun 6023   Fn wfn 6024  ⟶wf 6025  ‘cfv 6029   No csur 32123   <<s csslt 32226   |s cscut 32228   M cmade 32255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-wrecs 7557  df-recs 7619  df-1o 7711  df-2o 7712  df-no 32126  df-slt 32127  df-bday 32128  df-sslt 32227  df-scut 32229  df-made 32260

This theorem is referenced by:  madeval2  32266