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Theorem acfun 7027
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 2671 . . . 4 (𝜑𝐴 ∈ V)
3 abid2 2236 . . . . . 6 {𝑣𝑣𝑢} = 𝑢
4 vex 2661 . . . . . 6 𝑢 ∈ V
53, 4eqeltri 2188 . . . . 5 {𝑣𝑣𝑢} ∈ V
65a1i 9 . . . 4 ((𝜑𝑢𝐴) → {𝑣𝑣𝑢} ∈ V)
72, 6opabex3d 5985 . . 3 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V)
8 acfun.ac . . . 4 (𝜑CHOICE)
9 df-ac 7026 . . . 4 (CHOICE ↔ ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
108, 9sylib 121 . . 3 (𝜑 → ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
11 sseq2 3089 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓𝑦𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
12 dmeq 4707 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
1312fneq2d 5182 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
1411, 13anbi12d 462 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
1514exbidv 1779 . . . 4 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
1615spcgv 2745 . . 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 504 . . . . . 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 1674 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑤𝑥𝑤𝑢))
2120exbidv 1779 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤𝑢))
2221cbvralv 2629 . . . . . . . . . . 11 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑤 𝑤𝑢)
23 elequ1 1673 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
2423cbvexv 1870 . . . . . . . . . . . 12 (∃𝑤 𝑤𝑢 ↔ ∃𝑣 𝑣𝑢)
2524ralbii 2416 . . . . . . . . . . 11 (∀𝑢𝐴𝑤 𝑤𝑢 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2622, 25bitri 183 . . . . . . . . . 10 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2719, 26sylib 121 . . . . . . . . 9 (𝜑 → ∀𝑢𝐴𝑣 𝑣𝑢)
28 dmopab3 4720 . . . . . . . . 9 (∀𝑢𝐴𝑣 𝑣𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2927, 28sylib 121 . . . . . . . 8 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
3029fneq2d 5182 . . . . . . 7 (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3130adantr 272 . . . . . 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 507 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
34 fnopfv 5516 . . . . . . . . . 10 ((𝑓 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
3532, 34sylan 279 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
3633, 35sseldd 3066 . . . . . . . 8 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
37 vex 2661 . . . . . . . . 9 𝑥 ∈ V
38 vex 2661 . . . . . . . . . 10 𝑓 ∈ V
3938, 37fvex 5407 . . . . . . . . 9 (𝑓𝑥) ∈ V
40 eleq1 2178 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
41 elequ2 1674 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑣𝑢𝑣𝑥))
4240, 41anbi12d 462 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢𝐴𝑣𝑢) ↔ (𝑥𝐴𝑣𝑥)))
43 eleq1 2178 . . . . . . . . . 10 (𝑣 = (𝑓𝑥) → (𝑣𝑥 ↔ (𝑓𝑥) ∈ 𝑥))
4443anbi2d 457 . . . . . . . . 9 (𝑣 = (𝑓𝑥) → ((𝑥𝐴𝑣𝑥) ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥)))
4537, 39, 42, 44opelopab 4161 . . . . . . . 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 2480 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
4932, 48jca 302 . . . 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 1834 . 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 1312   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  Vcvv 2658  wss 3039  cop 3498  {copab 3956  dom cdm 4507   Fn wfn 5086  cfv 5091  CHOICEwac 7025
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ac 7026
This theorem is referenced by:  exmidaclem  7028
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