Theorem elold 28302

Description: Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
elold	⊢ (𝐴 ∈ On → (𝑋 ∈ (O‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))

Distinct variable groups:   𝐴,𝑏   𝑋,𝑏

Proof of Theorem elold

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oldval 28277	. . 3 ⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))
2	1	eleq2d 2850	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (O‘𝐴) ↔ 𝑋 ∈ ∪ (M “ 𝐴)))
3		eluni 4875	. . 3 ⊢ (𝑋 ∈ ∪ (M “ 𝐴) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)))
4		madef 28279	. . . . . . . 8 ⊢ M:On⟶𝒫 No
5		ffn 6739	. . . . . . . 8 ⊢ (M:On⟶𝒫 No → M Fn On)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ M Fn On
7		onss 7833	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8		fvelimab 6988	. . . . . . 7 ⊢ ((M Fn On ∧ 𝐴 ⊆ On) → (𝑦 ∈ (M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦))
9	6, 7, 8	sylancr 599	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑦 ∈ (M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦))
10	9	anbi2d 642	. . . . 5 ⊢ (𝐴 ∈ On → ((𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦)))
11	10	exbidv 1956	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦)))
12		fvex 6929	. . . . . . 7 ⊢ (M‘𝑏) ∈ V
13	12	clel3 3619	. . . . . 6 ⊢ (𝑋 ∈ (M‘𝑏) ↔ ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
14	13	rexbii 3113	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏) ↔ ∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
15		rexcom4 3293	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
16		eqcom 2770	. . . . . . . . 9 ⊢ (𝑦 = (M‘𝑏) ↔ (M‘𝑏) = 𝑦)
17	16	anbi2ci 637	. . . . . . . 8 ⊢ ((𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ (M‘𝑏) = 𝑦))
18	17	rexbii 3113	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ (M‘𝑏) = 𝑦))
19		r19.42v 3198	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ (M‘𝑏) = 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦))
20	18, 19	bitri 278	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦))
21	20	exbii 1881	. . . . 5 ⊢ (∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦))
22	14, 15, 21	3bitrri 301	. . . 4 ⊢ (∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏))
23	11, 22	bitrdi 290	. . 3 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))
24	3, 23	bitrid 286	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ∪ (M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))
25	2, 24	bitrd 282	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ (O‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4872   “ cima 5661  Oncon0 6388   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  Nocsur 28054  Mcmade 28265  Ocold 28266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271

This theorem is used by:  oldssmade  28310  oldlim  28330  madebdayim  28331  oldbdayim  28332  madebdaylemold  28341