Theorem wevgblacfn 35103

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 2818	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
2		raleq 3298	. . . . . . . . . . 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 3410	. . . . . . . . 9 ⊢ (𝑧 = ∅ → {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
5	4	unieqd 4887	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6		rab0 4352	. . . . . . . . . 10 ⊢ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
7	6	unieqi 4886	. . . . . . . . 9 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∪ ∅
8		uni0 4902	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
9	7, 8	eqtri 2753	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
10	5, 9	eqtrdi 2781	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11		0ex 5265	. . . . . . 7 ⊢ ∅ ∈ V
12	10, 11	eqeltrdi 2837	. . . . . 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 3974	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ V
15	14	jctl 523	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16		vex 3454	. . . . . . . . . . 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 5637	. . . . . . . . 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 4630	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ {𝑤}
23		eleq2 2818	. . . . . . . . . . . . 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 3657	. . . . . . . . . . . 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 4885	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑤})
28		unisnv 4894	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
29	27, 28	eqtrdi 2781	. . . . . . . . . . 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 1835	. . . . . . . . 9 ⊢ (∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32		reusn 4694	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33		df-rex 3055	. . . . . . . . 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 2817	. . . . . . . . 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 3127	. . . . . . 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 3474	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
41	13, 40	pm2.61dane 3013	. . . 4 ⊢ (𝑅 We V → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
42	41	ralrimivw 3130	. . 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 6661	. . 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 6982	. . . . . 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 2829	. . . 4 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐺‘𝑧) ∈ 𝑧)
49	48	ex 412	. . 3 ⊢ (𝑅 We V → (𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
50	49	alrimiv 1927	. 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191   We wwe 5593   Fn wfn 6509  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by: (None)