Theorem wevgblacfn 35225

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 2822	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
2		raleq 3290	. . . . . . . . . . 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 3398	. . . . . . . . 9 ⊢ (𝑧 = ∅ → {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
5	4	unieqd 4873	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6		rab0 4335	. . . . . . . . . 10 ⊢ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
7	6	unieqi 4872	. . . . . . . . 9 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∪ ∅
8		uni0 4888	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
9	7, 8	eqtri 2756	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
10	5, 9	eqtrdi 2784	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11		0ex 5249	. . . . . . 7 ⊢ ∅ ∈ V
12	10, 11	eqeltrdi 2841	. . . . . 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 3955	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ V
15	14	jctl 523	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16		vex 3441	. . . . . . . . . . 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 5617	. . . . . . . . 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 4617	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ {𝑤}
23		eleq2 2822	. . . . . . . . . . . . 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 3639	. . . . . . . . . . . 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 4871	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑤})
28		unisnv 4880	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
29	27, 28	eqtrdi 2784	. . . . . . . . . . 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 4681	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33		df-rex 3058	. . . . . . . . 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 2821	. . . . . . . . 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 3126	. . . . . . 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 3461	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
41	13, 40	pm2.61dane 3016	. . . 4 ⊢ (𝑅 We V → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
42	41	ralrimivw 3129	. . 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 6629	. . 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 6949	. . . . . 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 2833	. . . 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 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  ∃!wreu 3345  {crab 3396  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  {csn 4577  ∪ cuni 4860   class class class wbr 5095   ↦ cmpt 5176   We wwe 5573   Fn wfn 6484  ‘cfv 6489

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by: (None)