Theorem elold 28251

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 28226	. . 3 ⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))
2	1	eleq2d 2846	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ (O‘𝐴) ↔ 𝑋 ∈ ∪ (M “ 𝐴)))
3		eluni 4869	. . 3 ⊢ (𝑋 ∈ ∪ (M “ 𝐴) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)))
4		madef 28228	. . . . . . . 8 ⊢ M:On⟶𝒫 No
5		ffn 6710	. . . . . . . 8 ⊢ (M:On⟶𝒫 No → M Fn On)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ M Fn On
7		onss 7800	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
8		fvelimab 6958	. . . . . . 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 1953	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑦(𝑋 ∈ 𝑦 ∧ 𝑦 ∈ (M “ 𝐴)) ↔ ∃𝑦(𝑋 ∈ 𝑦 ∧ ∃𝑏 ∈ 𝐴 (M‘𝑏) = 𝑦)))
12		fvex 6899	. . . . . . 7 ⊢ (M‘𝑏) ∈ V
13	12	clel3 3615	. . . . . 6 ⊢ (𝑋 ∈ (M‘𝑏) ↔ ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
14	13	rexbii 3109	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑋 ∈ (M‘𝑏) ↔ ∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
15		rexcom4 3289	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 ∃𝑦(𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑦∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦))
16		eqcom 2767	. . . . . . . . 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 3109	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐴 (𝑦 = (M‘𝑏) ∧ 𝑋 ∈ 𝑦) ↔ ∃𝑏 ∈ 𝐴 (𝑋 ∈ 𝑦 ∧ (M‘𝑏) = 𝑦))
19		r19.42v 3194	. . . . . . 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 1880	. . . . 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 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866   “ cima 5653  Oncon0 6362   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  Nocsur 28003  Mcmade 28214  Ocold 28215

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 5326  ax-pr 5390  ax-un 7752

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219  df-old 28220

This theorem is used by:  oldssmade  28259  oldlim  28279  madebdayim  28280  oldbdayim  28281  madebdaylemold  28290