Theorem oldval 27801

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 27794	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8328	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6587	. . . . . . 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 6507	. . . . . 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 258	. . . . 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 6574	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
9	7, 8	mpan 690	. . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
10	9	uniexd 7681	. 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V)
11		imaeq2 6010	. . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
12	11	unieqd 4871	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴))
13		df-old 27795	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
14	12, 13	fvmptg 6933	. 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
15	10, 14	mpdan 687	1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  𝒫 cpw 4549  ∪ cuni 4858   ↦ cmpt 5174   × cxp 5617  ran crn 5620   “ cima 5622  Oncon0 6312  Fun wfun 6481   Fn wfn 6482  ‘cfv 6487  recscrecs 8296   |s cscut 27728   M cmade 27789   O cold 27790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-made 27794  df-old 27795

This theorem is referenced by:  old0  27806  elmade2  27819  elold  27820  old1  27826  oldss  27829  madeoldsuc  27836  oldfi  27865