Theorem madeoldsuc 33650

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 6179	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	imaeq2i 5903	. . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3		imaundi 5984	. . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
4	2, 3	eqtri 2781	. . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
5	4	unieqi 4814	. . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴}))
6		uniun 4826	. . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
7	5, 6	eqtri 2781	. . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})))
9		oldval 33624	. . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
10	9	eqcomd 2764	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴))
11		madef 33626	. . . . . . . 8 ⊢ M :On⟶𝒫 No
12		ffn 6502	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ M Fn On
14		fnsnfv 6735	. . . . . . . 8 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
15	14	eqcomd 2764	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
16	13, 15	mpan 689	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
17	16	unieqd 4815	. . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)})
18		fvex 6675	. . . . . 6 ⊢ ( M ‘𝐴) ∈ V
19	18	unisn 4823	. . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴)
20	17, 19	eqtrdi 2809	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴))
21	10, 20	uneq12d 4071	. . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22		oldssmade 33643	. . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
23		ssequn1 4087	. . . 4 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
24	22, 23	sylib 221	. . 3 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
25	8, 21, 24	3eqtrrd 2798	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴))
26		suceloni 7532	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
27		oldval 33624	. . 3 ⊢ (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴))
28	26, 27	syl 17	. 2 ⊢ (𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴))
29	25, 28	eqtr4d 2796	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3858   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   “ cima 5530  Oncon0 6173  suc csuc 6175   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339   No csur 33432   M cmade 33612   O cold 33613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-wrecs 7962  df-recs 8023  df-1o 8117  df-2o 8118  df-no 33435  df-slt 33436  df-bday 33437  df-sslt 33565  df-scut 33567  df-made 33617  df-old 33618

This theorem is referenced by:  oldsuc  33651  oldlim  33652