Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  acfun GIF version

Theorem acfun 7125
 Description: A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
Hypotheses
Ref Expression
acfun.ac (𝜑CHOICE)
acfun.a (𝜑𝐴𝑉)
acfun.m (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)
Assertion
Ref Expression
acfun (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
Distinct variable groups:   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥,𝑤
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝑉(𝑥,𝑤,𝑓)

Proof of Theorem acfun
Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 acfun.a . . . . 5 (𝜑𝐴𝑉)
21elexd 2725 . . . 4 (𝜑𝐴 ∈ V)
3 abid2 2278 . . . . . 6 {𝑣𝑣𝑢} = 𝑢
4 vex 2715 . . . . . 6 𝑢 ∈ V
53, 4eqeltri 2230 . . . . 5 {𝑣𝑣𝑢} ∈ V
65a1i 9 . . . 4 ((𝜑𝑢𝐴) → {𝑣𝑣𝑢} ∈ V)
72, 6opabex3d 6063 . . 3 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V)
8 acfun.ac . . . 4 (𝜑CHOICE)
9 df-ac 7124 . . . 4 (CHOICE ↔ ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
108, 9sylib 121 . . 3 (𝜑 → ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
11 sseq2 3152 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓𝑦𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
12 dmeq 4783 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
1312fneq2d 5258 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
1411, 13anbi12d 465 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
1514exbidv 1805 . . . 4 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
1615spcgv 2799 . . 3 ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V → (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
177, 10, 16sylc 62 . 2 (𝜑 → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
18 simprr 522 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
19 acfun.m . . . . . . . . . 10 (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)
20 elequ2 2133 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑤𝑥𝑤𝑢))
2120exbidv 1805 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤𝑢))
2221cbvralv 2680 . . . . . . . . . . 11 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑤 𝑤𝑢)
23 elequ1 2132 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
2423cbvexv 1898 . . . . . . . . . . . 12 (∃𝑤 𝑤𝑢 ↔ ∃𝑣 𝑣𝑢)
2524ralbii 2463 . . . . . . . . . . 11 (∀𝑢𝐴𝑤 𝑤𝑢 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2622, 25bitri 183 . . . . . . . . . 10 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2719, 26sylib 121 . . . . . . . . 9 (𝜑 → ∀𝑢𝐴𝑣 𝑣𝑢)
28 dmopab3 4796 . . . . . . . . 9 (∀𝑢𝐴𝑣 𝑣𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2927, 28sylib 121 . . . . . . . 8 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
3029fneq2d 5258 . . . . . . 7 (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3130adantr 274 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3218, 31mpbid 146 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn 𝐴)
33 simplrl 525 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
34 fnopfv 5594 . . . . . . . . . 10 ((𝑓 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
3532, 34sylan 281 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
3633, 35sseldd 3129 . . . . . . . 8 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
37 vex 2715 . . . . . . . . 9 𝑥 ∈ V
38 vex 2715 . . . . . . . . . 10 𝑓 ∈ V
3938, 37fvex 5485 . . . . . . . . 9 (𝑓𝑥) ∈ V
40 eleq1 2220 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
41 elequ2 2133 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑣𝑢𝑣𝑥))
4240, 41anbi12d 465 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢𝐴𝑣𝑢) ↔ (𝑥𝐴𝑣𝑥)))
43 eleq1 2220 . . . . . . . . . 10 (𝑣 = (𝑓𝑥) → (𝑣𝑥 ↔ (𝑓𝑥) ∈ 𝑥))
4443anbi2d 460 . . . . . . . . 9 (𝑣 = (𝑓𝑥) → ((𝑥𝐴𝑣𝑥) ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥)))
4537, 39, 42, 44opelopab 4230 . . . . . . . 8 (⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
4636, 45sylib 121 . . . . . . 7 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
4746simprd 113 . . . . . 6 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝑥)
4847ralrimiva 2530 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
4932, 48jca 304 . . . 4 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
5049ex 114 . . 3 (𝜑 → ((𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)))
5150eximdv 1860 . 2 (𝜑 → (∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)))
5217, 51mpd 13 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1333   = wceq 1335  ∃wex 1472   ∈ wcel 2128  {cab 2143  ∀wral 2435  Vcvv 2712   ⊆ wss 3102  ⟨cop 3563  {copab 4024  dom cdm 4583   Fn wfn 5162  ‘cfv 5167  CHOICEwac 7123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-ac 7124 This theorem is referenced by:  exmidaclem  7126
 Copyright terms: Public domain W3C validator