Theorem oldval 28250

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 28243	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8439	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6661	. . . . . . 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 6579	. . . . . 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 6648	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M “ 𝐴) ∈ V)
9	7, 8	mpan 703	. . 3 ⊢ (𝐴 ∈ On → (M “ 𝐴) ∈ V)
10	9	uniexd 7783	. 2 ⊢ (𝐴 ∈ On → ∪ (M “ 𝐴) ∈ V)
11		imaeq2 6060	. . . 4 ⊢ (𝑥 = 𝐴 → (M “ 𝑥) = (M “ 𝐴))
12	11	unieqd 4883	. . 3 ⊢ (𝑥 = 𝐴 → ∪ (M “ 𝑥) = ∪ (M “ 𝐴))
13		df-old 28244	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ (M “ 𝑥))
14	12, 13	fvmptg 7014	. 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 1570   ∈ wcel 2147  Vcvv 3454  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5189   × cxp 5653  ran crn 5656   “ cima 5658  Oncon0 6380  Fun wfun 6552   Fn wfn 6553  ‘cfv 6558  recscrecs 8406   |s ccuts 28175  Mcmade 28238  Ocold 28239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-made 28243  df-old 28244

This theorem is used by:  old0  28255  elmade2  28274  elold  28275  old1  28281  oldss  28286  madeoldsuc  28301  oldfi  28330  oldfib  28793