Theorem wevgblacfn 36080

Description: If 𝑅 is a well-ordering of the universe, then 𝐺 is a global choice function. Here 𝐺 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). This is the ZFC version of (3 → 1) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 29-Jun-2025.)

Hypothesis

Ref	Expression
wevgblacfn.1	⊢ 𝐺 = (𝑧 ∈ V ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})

Assertion

Ref	Expression
wevgblacfn	⊢ (𝑅 We V → (𝐺 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))

Distinct variable group:   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝐺(𝑥, 𝑦, 𝑧)

Proof of Theorem wevgblacfn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2853	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
2		raleq 3320	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦))
3	1, 2	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → ((𝑦 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦) ↔ (𝑦 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦)))
4	3	rabbidva2 3418	. . . . . . . . 9 ⊢ (𝑧 = ∅ → {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
5	4	unieqd 4885	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6		rab0 4338	. . . . . . . . . 10 ⊢ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
7	6	unieqi 4884	. . . . . . . . 9 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∪ ∅
8		uni0 4901	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
9	7, 8	eqtri 2786	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
10	5, 9	eqtrdi 2814	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11		0ex 5267	. . . . . . 7 ⊢ ∅ ∈ V
12	10, 11	eqeltrdi 2872	. . . . . 6 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
13	12	adantl 487	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 = ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
14		ssv 3937	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ V
15	14	jctl 533	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16		vex 3458	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
17	15, 16	jctil 529	. . . . . . . . . 10 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
18		3anass 1111	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
19	17, 18	sylibr 237	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
20		wereu 5654	. . . . . . . . 9 ⊢ ((𝑅 We V ∧ (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅)) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
21	19, 20	sylan2 605	. . . . . . . 8 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
22		vsnid 4625	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ {𝑤}
23		eleq2 2853	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ↔ 𝑤 ∈ {𝑤}))
24	22, 23	mpbiri 261	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
25		elrabi 3644	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} → 𝑤 ∈ 𝑧)
26	24, 25	syl 18	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ 𝑧)
27		unieq 4883	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑤})
28		unisnv 4892	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
29	27, 28	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
30	26, 29	jca 521	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
31	30	eximi 1868	. . . . . . . . 9 ⊢ (∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32		reusn 4690	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33		df-rex 3091	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
34	31, 32, 33	3imtr4i 295	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 → ∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
35	21, 34	syl 18	. . . . . . 7 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
36		eleq1 2852	. . . . . . . . 9 ⊢ (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
37	36	biimparc 485	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
38	37	rexlimiva 3159	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
39	35, 38	syl 18	. . . . . 6 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
40	39	elexd 3477	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
41	13, 40	pm2.61dane 3046	. . . 4 ⊢ (𝑅 We V → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
42	41	ralrimivw 3162	. . 3 ⊢ (𝑅 We V → ∀𝑧 ∈ V ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
43		wevgblacfn.1	. . . 4 ⊢ 𝐺 = (𝑧 ∈ V ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
44	43	fnmpt 6709	. . 3 ⊢ (∀𝑧 ∈ V ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V → 𝐺 Fn V)
45	42, 44	syl 18	. 2 ⊢ (𝑅 We V → 𝐺 Fn V)
46	43	fvmpt2 7036	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧) → (𝐺‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
47	16, 39, 46	sylancr 599	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐺‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
48	47, 39	eqeltrd 2864	. . . 4 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐺‘𝑧) ∈ 𝑧)
49	48	ex 418	. . 3 ⊢ (𝑅 We V → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
50	49	alrimiv 1962	. 2 ⊢ (𝑅 We V → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
51	45, 50	jca 521	1 ⊢ (𝑅 We V → (𝐺 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588  ∪ cuni 4872   class class class wbr 5108   ↦ cmpt 5192   We wwe 5610   Fn wfn 6561  ‘cfv 6566

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-fv 6574

This theorem is used by: (None)