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Theorem wevgblacfn 35114
Description: If 𝑅 is a well-ordering of the universe, then 𝐹 is a global choice function. Here 𝐹 maps each set 𝑧 to its minimal element with respect to 𝑅 (except when 𝑧 is the empty set, in which case it is mapped to the empty set, though this is only done for convenience). (Contributed by BTernaryTau, 29-Jun-2025.)
Hypothesis
Ref Expression
wevgblacfn.1 𝐹 = (𝑧 ∈ V ↦ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})
Assertion
Ref Expression
wevgblacfn (𝑅 We V → (𝐹 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧)))
Distinct variable group:   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem wevgblacfn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2830 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
2 raleq 3323 . . . . . . . . . . 11 (𝑧 = ∅ → (∀𝑥𝑧 ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦))
31, 2anbi12d 632 . . . . . . . . . 10 (𝑧 = ∅ → ((𝑦𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦) ↔ (𝑦 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦)))
43rabbidva2 3438 . . . . . . . . 9 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
54unieqd 4920 . . . . . . . 8 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6 rab0 4386 . . . . . . . . . 10 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
76unieqi 4919 . . . . . . . . 9 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} =
8 uni0 4935 . . . . . . . . 9 ∅ = ∅
97, 8eqtri 2765 . . . . . . . 8 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
105, 9eqtrdi 2793 . . . . . . 7 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11 0ex 5307 . . . . . . 7 ∅ ∈ V
1210, 11eqeltrdi 2849 . . . . . 6 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
1312adantl 481 . . . . 5 ((𝑅 We V ∧ 𝑧 = ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
14 ssv 4008 . . . . . . . . . . . 12 𝑧 ⊆ V
1514jctl 523 . . . . . . . . . . 11 (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16 vex 3484 . . . . . . . . . . 11 𝑧 ∈ V
1715, 16jctil 519 . . . . . . . . . 10 (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
18 3anass 1095 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
1917, 18sylibr 234 . . . . . . . . 9 (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
20 wereu 5681 . . . . . . . . 9 ((𝑅 We V ∧ (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅)) → ∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦)
2119, 20sylan2 593 . . . . . . . 8 ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦)
22 vsnid 4663 . . . . . . . . . . . . 13 𝑤 ∈ {𝑤}
23 eleq2 2830 . . . . . . . . . . . . 13 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ↔ 𝑤 ∈ {𝑤}))
2422, 23mpbiri 258 . . . . . . . . . . . 12 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})
25 elrabi 3687 . . . . . . . . . . . 12 (𝑤 ∈ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} → 𝑤𝑧)
2624, 25syl 17 . . . . . . . . . . 11 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤𝑧)
27 unieq 4918 . . . . . . . . . . . 12 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
28 unisnv 4927 . . . . . . . . . . . 12 {𝑤} = 𝑤
2927, 28eqtrdi 2793 . . . . . . . . . . 11 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
3026, 29jca 511 . . . . . . . . . 10 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
3130eximi 1835 . . . . . . . . 9 (∃𝑤{𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32 reusn 4727 . . . . . . . . 9 (∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33 df-rex 3071 . . . . . . . . 9 (∃𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 ↔ ∃𝑤(𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
3431, 32, 333imtr4i 292 . . . . . . . 8 (∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦 → ∃𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
3521, 34syl 17 . . . . . . 7 ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
36 eleq1 2829 . . . . . . . . 9 ( {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → ( {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧𝑤𝑧))
3736biimparc 479 . . . . . . . 8 ((𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
3837rexlimiva 3147 . . . . . . 7 (∃𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
3935, 38syl 17 . . . . . 6 ((𝑅 We V ∧ 𝑧 ≠ ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
4039elexd 3504 . . . . 5 ((𝑅 We V ∧ 𝑧 ≠ ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
4113, 40pm2.61dane 3029 . . . 4 (𝑅 We V → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
4241ralrimivw 3150 . . 3 (𝑅 We V → ∀𝑧 ∈ V {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
43 wevgblacfn.1 . . . 4 𝐹 = (𝑧 ∈ V ↦ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})
4443fnmpt 6708 . . 3 (∀𝑧 ∈ V {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V → 𝐹 Fn V)
4542, 44syl 17 . 2 (𝑅 We V → 𝐹 Fn V)
4643fvmpt2 7027 . . . . . 6 ((𝑧 ∈ V ∧ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧) → (𝐹𝑧) = {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})
4716, 39, 46sylancr 587 . . . . 5 ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐹𝑧) = {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦})
4847, 39eqeltrd 2841 . . . 4 ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐹𝑧) ∈ 𝑧)
4948ex 412 . . 3 (𝑅 We V → (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
5049alrimiv 1927 . 2 (𝑅 We V → ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
5145, 50jca 511 1 (𝑅 We V → (𝐹 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  ∃!wreu 3378  {crab 3436  Vcvv 3480  wss 3951  c0 4333  {csn 4626   cuni 4907   class class class wbr 5143  cmpt 5225   We wwe 5636   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by: (None)
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