Theorem oldval 28221

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 28214	. . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))
2		recsfnon 8412	. . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On
3		fnfun 6640	. . . . . . 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 6560	. . . . . 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 6627	. . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → (M “ 𝐴) ∈ V)
9	7, 8	mpan 703	. . 3 ⊢ (𝐴 ∈ On → (M “ 𝐴) ∈ V)
10	9	uniexd 7760	. 2 ⊢ (𝐴 ∈ On → ∪ (M “ 𝐴) ∈ V)
11		imaeq2 6048	. . . 4 ⊢ (𝑥 = 𝐴 → (M “ 𝑥) = (M “ 𝐴))
12	11	unieqd 4880	. . 3 ⊢ (𝑥 = 𝐴 → ∪ (M “ 𝑥) = ∪ (M “ 𝐴))
13		df-old 28215	. . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ (M “ 𝑥))
14	12, 13	fvmptg 6992	. 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 2145  Vcvv 3451  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5649  ran crn 5652   “ cima 5654  Oncon0 6363  Fun wfun 6533   Fn wfn 6534  ‘cfv 6539  recscrecs 8379   |s ccuts 28146  Mcmade 28209  Ocold 28210

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7752

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-2nd 8003  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-made 28214  df-old 28215

This theorem is used by:  old0  28226  elmade2  28245  elold  28246  old1  28252  oldss  28257  madeoldsuc  28272  oldfi  28301  oldfib  28764