Theorem madeoldsuc 27772

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 6326	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	imaeq2i 6018	. . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴}))
3		imaundi 6110	. . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
4	2, 3	eqtri 2752	. . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴}))
5	4	unieqi 4879	. . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴}))
6		uniun 4890	. . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
7	5, 6	eqtri 2752	. . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})))
9		oldval 27738	. . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
10	9	eqcomd 2735	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴))
11		madef 27740	. . . . . . . 8 ⊢ M :On⟶𝒫 No
12		ffn 6670	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ M Fn On
14		fnsnfv 6922	. . . . . . . 8 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴}))
15	14	eqcomd 2735	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)})
16	13, 15	mpan 690	. . . . . 6 ⊢ (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)})
17	16	unieqd 4880	. . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)})
18		fvex 6853	. . . . . 6 ⊢ ( M ‘𝐴) ∈ V
19	18	unisn 4886	. . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴)
20	17, 19	eqtrdi 2780	. . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴))
21	10, 20	uneq12d 4128	. . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴)))
22		oldssmade 27765	. . . . 5 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴)
23	22	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴))
24		ssequn1 4145	. . . 4 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
25	23, 24	sylib 218	. . 3 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴))
26	8, 21, 25	3eqtrrd 2769	. 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴))
27		onsuc 7767	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
28		oldval 27738	. . 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 2767	1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3909   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867   “ cima 5634  Oncon0 6320  suc csuc 6322   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499   No csur 27527   M cmade 27726   O cold 27727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-no 27530  df-slt 27531  df-bday 27532  df-sslt 27669  df-scut 27671  df-made 27731  df-old 27732

This theorem is referenced by:  oldsuc  27773  oldlim  27774