Theorem oldval 28220

Description: The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
oldval	⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))

Proof of Theorem oldval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-made 28213	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8411	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6639	. . . . . . 7 ⊢ (recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On → Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))))
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
5		funeq 6559	. . . . . 6 ⊢ (M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) → (Fun M ↔ Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))))
6	4, 5	mpbiri 261	. . . . 5 ⊢ (M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) → Fun M)
7	1, 6	ax-mp 5	. . . 4 ⊢ Fun M
8		funimaexg 6626	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M “ 𝐴) ∈ V)
9	7, 8	mpan 703	. . 3 ⊢ (𝐴 ∈ On → (M “ 𝐴) ∈ V)
10	9	uniexd 7759	. 2 ⊢ (𝐴 ∈ On → ∪ (M “ 𝐴) ∈ V)
11		imaeq2 6047	. . . 4 ⊢ (𝑥 = 𝐴 → (M “ 𝑥) = (M “ 𝐴))
12	11	unieqd 4879	. . 3 ⊢ (𝑥 = 𝐴 → ∪ (M “ 𝑥) = ∪ (M “ 𝐴))
13		df-old 28214	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ (M “ 𝑥))
14	12, 13	fvmptg 6991	. 2 ⊢ ((𝐴 ∈ On ∧ ∪ (M “ 𝐴) ∈ V) → (O‘𝐴) = ∪ (M “ 𝐴))
15	10, 14	mpdan 700	1 ⊢ (𝐴 ∈ On → (O‘𝐴) = ∪ (M “ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  𝒫 cpw 4556  ∪ cuni 4866   ↦ cmpt 5185   × cxp 5648  ran crn 5651   “ cima 5653  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  recscrecs 8378   |s ccuts 28145  Mcmade 28208  Ocold 28209

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-pr 5390  ax-un 7751

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-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 4584  df-pr 4586  df-op 4590  df-uni 4867  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 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-made 28213  df-old 28214

This theorem is used by:  old0  28225  elmade2  28244  elold  28245  old1  28251  oldss  28256  madeoldsuc  28271  oldfi  28300  oldfib  28763