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Theorem wevgblacfn 35092
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 2827 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧𝑦 ∈ ∅))
2 raleq 3320 . . . . . . . . . . 11 (𝑧 = ∅ → (∀𝑥𝑧 ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦))
31, 2anbi12d 632 . . . . . . . . . 10 (𝑧 = ∅ → ((𝑦𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦) ↔ (𝑦 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦)))
43rabbidva2 3434 . . . . . . . . 9 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
54unieqd 4924 . . . . . . . 8 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6 rab0 4391 . . . . . . . . . 10 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
76unieqi 4923 . . . . . . . . 9 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} =
8 uni0 4939 . . . . . . . . 9 ∅ = ∅
97, 8eqtri 2762 . . . . . . . 8 {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
105, 9eqtrdi 2790 . . . . . . 7 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11 0ex 5312 . . . . . . 7 ∅ ∈ V
1210, 11eqeltrdi 2846 . . . . . 6 (𝑧 = ∅ → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
1312adantl 481 . . . . 5 ((𝑅 We V ∧ 𝑧 = ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
14 ssv 4019 . . . . . . . . . . . 12 𝑧 ⊆ V
1514jctl 523 . . . . . . . . . . 11 (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16 vex 3481 . . . . . . . . . . 11 𝑧 ∈ V
1715, 16jctil 519 . . . . . . . . . 10 (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
18 3anass 1094 . . . . . . . . . 10 ((𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
1917, 18sylibr 234 . . . . . . . . 9 (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
20 wereu 5684 . . . . . . . . 9 ((𝑅 We V ∧ (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅)) → ∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦)
2119, 20sylan2 593 . . . . . . . 8 ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦)
22 vsnid 4667 . . . . . . . . . . . . 13 𝑤 ∈ {𝑤}
23 eleq2 2827 . . . . . . . . . . . . 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 3689 . . . . . . . . . . . 12 (𝑤 ∈ {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} → 𝑤𝑧)
2624, 25syl 17 . . . . . . . . . . 11 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤𝑧)
27 unieq 4922 . . . . . . . . . . . 12 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
28 unisnv 4931 . . . . . . . . . . . 12 {𝑤} = 𝑤
2927, 28eqtrdi 2790 . . . . . . . . . . 11 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
3026, 29jca 511 . . . . . . . . . 10 ({𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
3130eximi 1831 . . . . . . . . 9 (∃𝑤{𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32 reusn 4731 . . . . . . . . 9 (∃!𝑦𝑧𝑥𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33 df-rex 3068 . . . . . . . . 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 2826 . . . . . . . . 9 ( {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → ( {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧𝑤𝑧))
3736biimparc 479 . . . . . . . 8 ((𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
3837rexlimiva 3144 . . . . . . 7 (∃𝑤𝑧 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
3935, 38syl 17 . . . . . 6 ((𝑅 We V ∧ 𝑧 ≠ ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
4039elexd 3501 . . . . 5 ((𝑅 We V ∧ 𝑧 ≠ ∅) → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
4113, 40pm2.61dane 3026 . . . 4 (𝑅 We V → {𝑦𝑧 ∣ ∀𝑥𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
4241ralrimivw 3147 . . 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 7026 . . . . . 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 2838 . . . 4 ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐹𝑧) ∈ 𝑧)
4948ex 412 . . 3 (𝑅 We V → (𝑧 ≠ ∅ → (𝐹𝑧) ∈ 𝑧))
5049alrimiv 1924 . 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 1086  wal 1534   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  ∃!wreu 3375  {crab 3432  Vcvv 3477  wss 3962  c0 4338  {csn 4630   cuni 4911   class class class wbr 5147  cmpt 5230   We wwe 5639   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by: (None)
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