Theorem elold 33980

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 33965	. . 3 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
2	1	eleq2d 2824	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ 𝑋 ∈ ∪ ( M “ 𝐴)))
3		eluni 4839	. . 3 ⊢ (𝑋 ∈ ∪ ( M “ 𝐴) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)))
4		madef 33967	. . . . . . . 8 ⊢ M :On⟶𝒫 No
5		ffn 6584	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ M Fn On
7		onss 7611	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8		fvelimab 6823	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ⊆ On) → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
9	6, 7, 8	sylancr 586	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
10	9	anbi2d 628	. . . . 5 ⊢ (𝐴 ∈ On → ((𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
11	10	exbidv 1925	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
12		fvex 6769	. . . . . . 7 ⊢ ( M ‘𝑏) ∈ V
13	12	clel3 3585	. . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
14	13	rexbii 3177	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
15		rexcom4 3179	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
16		eqcom 2745	. . . . . . . . 9 ⊢ (𝑦 = ( M ‘𝑏) ↔ ( M ‘𝑏) = 𝑦)
17	16	anbi2ci 624	. . . . . . . 8 ⊢ ((𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
18	17	rexbii 3177	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
19		r19.42v 3276	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
20	18, 19	bitri 274	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
21	20	exbii 1851	. . . . 5 ⊢ (∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
22	14, 15, 21	3bitrri 297	. . . 4 ⊢ (∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))
23	11, 22	bitrdi 286	. . 3 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
24	3, 23	syl5bb 282	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ∪ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
25	2, 24	bitrd 278	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   “ cima 5583  Oncon0 6251   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418   No csur 33770   M cmade 33953   O cold 33954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959

This theorem is referenced by:  oldssmade  33987  oldlim  33996  madebdayim  33997  oldbdayim  33998  madebdaylemold  34005