Theorem oldval 27826

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 27819	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8342	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6599	. . . . . . 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 6519	. . . . . 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 6586	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
9	7, 8	mpan 691	. . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
10	9	uniexd 7696	. 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V)
11		imaeq2 6022	. . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
12	11	unieqd 4864	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴))
13		df-old 27820	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
14	12, 13	fvmptg 6946	. 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴))
15	10, 14	mpdan 688	1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  𝒫 cpw 4542  ∪ cuni 4851   ↦ cmpt 5167   × cxp 5629  ran crn 5632   “ cima 5634  Oncon0 6324  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  recscrecs 8310   |s ccuts 27751   M cmade 27814   O cold 27815

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 5213  ax-sep 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

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-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-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-made 27819  df-old 27820

This theorem is referenced by:  old0  27831  elmade2  27850  elold  27851  old1  27857  oldss  27862  madeoldsuc  27877  oldfi  27906  oldfib  28369