Theorem madeval 28224

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 28219	. . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2	1	tfr2 8410	. 2 ⊢ (𝐴 ∈ On → (M‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘(M ↾ 𝐴)))
3		eqid 2760	. . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))
4		rneq 5917	. . . . . . . 8 ⊢ (𝑥 = (M ↾ 𝐴) → ran 𝑥 = ran (M ↾ 𝐴))
5		df-ima 5663	. . . . . . . 8 ⊢ (M “ 𝐴) = ran (M ↾ 𝐴)
6	4, 5	eqtr4di 2813	. . . . . . 7 ⊢ (𝑥 = (M ↾ 𝐴) → ran 𝑥 = (M “ 𝐴))
7	6	unieqd 4879	. . . . . 6 ⊢ (𝑥 = (M ↾ 𝐴) → ∪ ran 𝑥 = ∪ (M “ 𝐴))
8	7	pweqd 4573	. . . . 5 ⊢ (𝑥 = (M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ (M “ 𝐴))
9	8	sqxpeqd 5682	. . . 4 ⊢ (𝑥 = (M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)))
10	9	imaeq2d 6051	. . 3 ⊢ (𝑥 = (M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))))
11	1	tfr1 8409	. . . . 5 ⊢ M Fn On
12		fnfun 6640	. . . . 5 ⊢ (M Fn On → Fun M)
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun M
14		resfunexg 7222	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M ↾ 𝐴) ∈ V)
15	13, 14	mpan 703	. . 3 ⊢ (𝐴 ∈ On → (M ↾ 𝐴) ∈ V)
16		cutsf 28184	. . . . 5 ⊢ |s : <<s ⟶No
17		ffun 6713	. . . . 5 ⊢ ( |s : <<s ⟶No → Fun |s )
18	16, 17	ax-mp 5	. . . 4 ⊢ Fun |s
19		funimaexg 6627	. . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M “ 𝐴) ∈ V)
20	13, 19	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ On → (M “ 𝐴) ∈ V)
21		uniexg 7758	. . . . . 6 ⊢ ((M “ 𝐴) ∈ V → ∪ (M “ 𝐴) ∈ V)
22		pwexg 5339	. . . . . 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 7766	. . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴)) ∈ V)
25		funimaexg 6627	. . . 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 7018	. 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 2795	1 ⊢ (𝐴 ∈ On → (M‘𝐴) = ( |s “ (𝒫 ∪ (M “ 𝐴) × 𝒫 ∪ (M “ 𝐴))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  𝒫 cpw 4556  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5648  ran crn 5651   ↾ cres 5652   “ cima 5653  Oncon0 6362  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  Nocsur 28003   <<s cslts 28149   |s ccuts 28151  Mcmade 28214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219

This theorem is used by:  madeval2  28225  madefi  28305