Theorem wevgblacfn 35145

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 2820	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
2		raleq 3289	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦))
3	1, 2	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → ((𝑦 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦) ↔ (𝑦 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦)))
4	3	rabbidva2 3397	. . . . . . . . 9 ⊢ (𝑧 = ∅ → {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
5	4	unieqd 4867	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6		rab0 4331	. . . . . . . . . 10 ⊢ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
7	6	unieqi 4866	. . . . . . . . 9 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∪ ∅
8		uni0 4882	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
9	7, 8	eqtri 2754	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
10	5, 9	eqtrdi 2782	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11		0ex 5240	. . . . . . 7 ⊢ ∅ ∈ V
12	10, 11	eqeltrdi 2839	. . . . . 6 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
13	12	adantl 481	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 = ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
14		ssv 3954	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ V
15	14	jctl 523	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16		vex 3440	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
17	15, 16	jctil 519	. . . . . . . . . 10 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
18		3anass 1094	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
19	17, 18	sylibr 234	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
20		wereu 5607	. . . . . . . . 9 ⊢ ((𝑅 We V ∧ (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅)) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
21	19, 20	sylan2 593	. . . . . . . 8 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
22		vsnid 4611	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ {𝑤}
23		eleq2 2820	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ↔ 𝑤 ∈ {𝑤}))
24	22, 23	mpbiri 258	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
25		elrabi 3638	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} → 𝑤 ∈ 𝑧)
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ 𝑧)
27		unieq 4865	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑤})
28		unisnv 4874	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
29	27, 28	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
30	26, 29	jca 511	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
31	30	eximi 1836	. . . . . . . . 9 ⊢ (∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32		reusn 4675	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33		df-rex 3057	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
34	31, 32, 33	3imtr4i 292	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 → ∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
35	21, 34	syl 17	. . . . . . 7 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤)
36		eleq1 2819	. . . . . . . . 9 ⊢ (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
37	36	biimparc 479	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
38	37	rexlimiva 3125	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
39	35, 38	syl 17	. . . . . 6 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
40	39	elexd 3460	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
41	13, 40	pm2.61dane 3015	. . . 4 ⊢ (𝑅 We V → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
42	41	ralrimivw 3128	. . 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 6616	. . 3 ⊢ (∀𝑧 ∈ V ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V → 𝐺 Fn V)
45	42, 44	syl 17	. 2 ⊢ (𝑅 We V → 𝐺 Fn V)
46	43	fvmpt2 6935	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧) → (𝐺‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
47	16, 39, 46	sylancr 587	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐺‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
48	47, 39	eqeltrd 2831	. . . 4 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐺‘𝑧) ∈ 𝑧)
49	48	ex 412	. . 3 ⊢ (𝑅 We V → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
50	49	alrimiv 1928	. 2 ⊢ (𝑅 We V → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
51	45, 50	jca 511	1 ⊢ (𝑅 We V → (𝐺 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  ∃!wreu 3344  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4278  {csn 4571  ∪ cuni 4854   class class class wbr 5086   ↦ cmpt 5167   We wwe 5563   Fn wfn 6471  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484

This theorem is referenced by: (None)