Theorem madeoldsuc 34067

Description: The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
madeoldsuc	⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Proof of Theorem madeoldsuc

Step	Hyp	Ref	Expression
1		df-suc 6272	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	imaeq2i 5967	. . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3		imaundi 6053	. . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
4	2, 3	eqtri 2766	. . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
5	4	unieqi 4852	. . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴}))
6		uniun 4864	. . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
7	5, 6	eqtri 2766	. . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})))
9		oldval 34038	. . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
10	9	eqcomd 2744	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴))
11		madef 34040	. . . . . . . 8 ⊢ M :On⟶𝒫 No
12		ffn 6600	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ M Fn On
14		fnsnfv 6847	. . . . . . . 8 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
15	14	eqcomd 2744	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
16	13, 15	mpan 687	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
17	16	unieqd 4853	. . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)})
18		fvex 6787	. . . . . 6 ⊢ ( M ‘𝐴) ∈ V
19	18	unisn 4861	. . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴)
20	17, 19	eqtrdi 2794	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴))
21	10, 20	uneq12d 4098	. . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22		oldssmade 34060	. . . . 5 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
23	22	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
24		ssequn1 4114	. . . 4 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
25	23, 24	sylib 217	. . 3 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
26	8, 21, 25	3eqtrrd 2783	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴))
27		suceloni 7659	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
28		oldval 34038	. . 3 ⊢ (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴))
29	27, 28	syl 17	. 2 ⊢ (𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴))
30	26, 29	eqtr4d 2781	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   “ cima 5592  Oncon0 6266  suc csuc 6268   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433   No csur 33843   M cmade 34026   O cold 34027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-made 34031  df-old 34032

This theorem is referenced by:  oldsuc  34068  oldlim  34069