Theorem madeoldsuc 28280

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 6371	. . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	imaeq2i 6052	. . . . . . 7 ⊢ (M “ suc 𝐴) = (M “ (𝐴 ∪ {𝐴}))
3		imaundi 6143	. . . . . . 7 ⊢ (M “ (𝐴 ∪ {𝐴})) = ((M “ 𝐴) ∪ (M “ {𝐴}))
4	2, 3	eqtri 2783	. . . . . 6 ⊢ (M “ suc 𝐴) = ((M “ 𝐴) ∪ (M “ {𝐴}))
5	4	unieqi 4878	. . . . 5 ⊢ ∪ (M “ suc 𝐴) = ∪ ((M “ 𝐴) ∪ (M “ {𝐴}))
6		uniun 4889	. . . . 5 ⊢ ∪ ((M “ 𝐴) ∪ (M “ {𝐴})) = (∪ (M “ 𝐴) ∪ ∪ (M “ {𝐴}))
7	5, 6	eqtri 2783	. . . 4 ⊢ ∪ (M “ suc 𝐴) = (∪ (M “ 𝐴) ∪ ∪ (M “ {𝐴}))
8	7	a1i 11	. . 3 ⊢ (𝐴 ∈ On → ∪ (M “ suc 𝐴) = (∪ (M “ 𝐴) ∪ ∪ (M “ {𝐴})))
9		oldval 28229	. . . . 5 ⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))
10	9	eqcomd 2766	. . . 4 ⊢ (𝐴 ∈ On → ∪ (M “ 𝐴) = (O‘𝐴))
11		madef 28231	. . . . . . . 8 ⊢ M:On⟶𝒫 No
12		ffn 6714	. . . . . . . 8 ⊢ (M:On⟶𝒫 No → M Fn On)
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ M Fn On
14		fnsnfv 6969	. . . . . . . 8 ⊢ ((M Fn On ∧ 𝐴 ∈ On) → {(M‘𝐴)} = (M “ {𝐴}))
15	14	eqcomd 2766	. . . . . . 7 ⊢ ((M Fn On ∧ 𝐴 ∈ On) → (M “ {𝐴}) = {(M‘𝐴)})
16	13, 15	mpan 703	. . . . . 6 ⊢ (𝐴 ∈ On → (M “ {𝐴}) = {(M‘𝐴)})
17	16	unieqd 4879	. . . . 5 ⊢ (𝐴 ∈ On → ∪ (M “ {𝐴}) = ∪ {(M‘𝐴)})
18		fvex 6903	. . . . . 6 ⊢ (M‘𝐴) ∈ V
19	18	unisn 4885	. . . . 5 ⊢ ∪ {(M‘𝐴)} = (M‘𝐴)
20	17, 19	eqtrdi 2811	. . . 4 ⊢ (𝐴 ∈ On → ∪ (M “ {𝐴}) = (M‘𝐴))
21	10, 20	uneq12d 4167	. . 3 ⊢ (𝐴 ∈ On → (∪ (M “ 𝐴) ∪ ∪ (M “ {𝐴})) = ((O‘𝐴) ∪ (M‘𝐴)))
22		oldssmade 28262	. . . . 5 ⊢ (O‘𝐴) ⊆ (M‘𝐴)
23	22	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → (O‘𝐴) ⊆ (M‘𝐴))
24		ssequn1 4183	. . . 4 ⊢ ((O‘𝐴) ⊆ (M‘𝐴) ↔ ((O‘𝐴) ∪ (M‘𝐴)) = (M‘𝐴))
25	23, 24	sylib 221	. . 3 ⊢ (𝐴 ∈ On → ((O‘𝐴) ∪ (M‘𝐴)) = (M‘𝐴))
26	8, 21, 25	3eqtrrd 2800	. 2 ⊢ (𝐴 ∈ On → (M‘𝐴) = ∪ (M “ suc 𝐴))
27		onsuc 7828	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
28		oldval 28229	. . 3 ⊢ (suc 𝐴 ∈ On → (O‘suc 𝐴) = ∪ (M “ suc 𝐴))
29	27, 28	syl 18	. 2 ⊢ (𝐴 ∈ On → (O‘suc 𝐴) = ∪ (M “ suc 𝐴))
30	26, 29	eqtr4d 2798	1 ⊢ (𝐴 ∈ On → (M‘𝐴) = (O‘suc 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∪ cun 4148  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866   “ cima 5654  Oncon0 6365  suc csuc 6367   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  Nocsur 28006  Mcmade 28217  Ocold 28218

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 5327  ax-pr 5391  ax-un 7755

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-1o 8483  df-2o 8484  df-no 28009  df-lts 28010  df-bday 28011  df-slts 28153  df-cuts 28155  df-made 28222  df-old 28223

This theorem is used by:  oldsuc  28281  oldlim  28282