Theorem elold 27865

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 27840	. . 3 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
2	1	eleq2d 2823	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ 𝑋 ∈ ∪ ( M “ 𝐴)))
3		eluni 4854	. . 3 ⊢ (𝑋 ∈ ∪ ( M “ 𝐴) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)))
4		madef 27842	. . . . . . . 8 ⊢ M :On⟶𝒫 No
5		ffn 6662	. . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ M Fn On
7		onss 7732	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8		fvelimab 6906	. . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ⊆ On) → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
9	6, 7, 8	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑦 ∈ ( M “ 𝐴) ↔ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦))
10	9	anbi2d 631	. . . . 5 ⊢ (𝐴 ∈ On → ((𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ (𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
11	10	exbidv 1923	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴)) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 ( M ‘𝑏) = 𝑦)))
12		fvex 6847	. . . . . . 7 ⊢ ( M ‘𝑏) ∈ V
13	12	clel3 3605	. . . . . 6 ⊢ (𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
14	13	rexbii 3085	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏) ↔ ∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
15		rexcom4 3265	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦))
16		eqcom 2744	. . . . . . . . 9 ⊢ (𝑦 = ( M ‘𝑏) ↔ ( M ‘𝑏) = 𝑦)
17	16	anbi2ci 626	. . . . . . . 8 ⊢ ((𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
18	17	rexbii 3085	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = ( M ‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ ( M ‘𝑏) = 𝑦))
19		r19.42v 3170	. . . . . . 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 1850	. . . . 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   “ cima 5627  Oncon0 6317   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492   No csur 27617   M cmade 27828   O cold 27829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-made 27833  df-old 27834

This theorem is referenced by:  oldssmade  27873  oldlim  27893  madebdayim  27894  oldbdayim  27895  madebdaylemold  27904