Theorem elold 27923

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 27908	. . 3 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
2	1	eleq2d 2825	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ 𝑋 ∈ ∪ ( M “ 𝐴)))
3		eluni 4915	. . 3 ⊢ (𝑋 ∈ ∪ ( M “ 𝐴) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)))
4		madef 27910	. . . . . . . 8 ⊢ M :On⟶𝒫 No
5		ffn 6737	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ M Fn On
7		onss 7804	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8		fvelimab 6981	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ⊆ On) → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
9	6, 7, 8	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
10	9	anbi2d 630	. . . . 5 ⊢ (𝐴 ∈ On → ((𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
11	10	exbidv 1919	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
12		fvex 6920	. . . . . . 7 ⊢ ( M ‘𝑏) ∈ V
13	12	clel3 3662	. . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
14	13	rexbii 3092	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
15		rexcom4 3286	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
16		eqcom 2742	. . . . . . . . 9 ⊢ (𝑦 = ( M ‘𝑏) ↔ ( M ‘𝑏) = 𝑦)
17	16	anbi2ci 625	. . . . . . . 8 ⊢ ((𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
18	17	rexbii 3092	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
19		r19.42v 3189	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
20	18, 19	bitri 275	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
21	20	exbii 1845	. . . . 5 ⊢ (∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
22	14, 15, 21	3bitrri 298	. . . 4 ⊢ (∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏))
23	11, 22	bitrdi 287	. . 3 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
24	3, 23	bitrid 283	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ∪ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
25	2, 24	bitrd 279	1 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∃wrex 3068   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912   “ cima 5692  Oncon0 6386   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563   No csur 27699   M cmade 27896   O cold 27897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902

This theorem is referenced by:  oldssmade  27931  oldlim  27940  madebdayim  27941  oldbdayim  27942  madebdaylemold  27951