Theorem oldval 27814

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 27807	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8333	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6590	. . . . . . 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 6510	. . . . . 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 6577	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V)
9	7, 8	mpan 691	. . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V)
10	9	uniexd 7687	. 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V)
11		imaeq2 6013	. . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴))
12	11	unieqd 4864	. . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴))
13		df-old 27808	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥))
14	12, 13	fvmptg 6937	. 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 5620  ran crn 5623   “ cima 5625  Oncon0 6315  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490  recscrecs 8301   |s ccuts 27739   M cmade 27802   O cold 27803

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-pr 5368  ax-un 7680

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 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-made 27807  df-old 27808

This theorem is referenced by:  old0  27819  elmade2  27838  elold  27839  old1  27845  oldss  27850  madeoldsuc  27865  oldfi  27894  oldfib  28357