Theorem madeoldsuc 27895

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 6316	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	imaeq2i 6010	. . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3		imaundi 6100	. . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
4	2, 3	eqtri 2762	. . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
5	4	unieqi 4850	. . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴}))
6		uniun 4861	. . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
7	5, 6	eqtri 2762	. . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})))
9		oldval 27844	. . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
10	9	eqcomd 2745	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴))
11		madef 27846	. . . . . . . 8 ⊢ M :On⟶𝒫 No
12		ffn 6655	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ M Fn On
14		fnsnfv 6906	. . . . . . . 8 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
15	14	eqcomd 2745	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
16	13, 15	mpan 696	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
17	16	unieqd 4851	. . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)})
18		fvex 6840	. . . . . 6 ⊢ ( M ‘𝐴) ∈ V
19	18	unisn 4857	. . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴)
20	17, 19	eqtrdi 2790	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴))
21	10, 20	uneq12d 4099	. . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22		oldssmade 27877	. . . . 5 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
23	22	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
24		ssequn1 4115	. . . 4 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
25	23, 24	sylib 219	. . 3 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
26	8, 21, 25	3eqtrrd 2779	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴))
27		onsuc 7753	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
28		oldval 27844	. . 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 2777	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838   “ cima 5621  Oncon0 6310  suc csuc 6312   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485   No csur 27621   M cmade 27832   O cold 27833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-no 27624  df-lts 27625  df-bday 27626  df-slts 27768  df-cuts 27770  df-made 27837  df-old 27838

This theorem is referenced by:  oldsuc  27896  oldlim  27897