Theorem wevgblacfn 34946

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 ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})

Assertion

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

Step	Hyp	Ref	Expression
1		eleq2 2815	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ ∅))
2		raleq 3312	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦))
3	1, 2	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → ((𝑦 ∈ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦) ↔ (𝑦 ∈ ∅ ∧ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦)))
4	3	rabbidva2 3421	. . . . . . . . 9 ⊢ (𝑧 = ∅ → {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
5	4	unieqd 4918	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦})
6		rab0 4380	. . . . . . . . . 10 ⊢ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
7	6	unieqi 4917	. . . . . . . . 9 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∪ ∅
8		uni0 4935	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
9	7, 8	eqtri 2754	. . . . . . . 8 ⊢ ∪ {𝑦 ∈ ∅ ∣ ∀𝑥 ∈ ∅ ¬ 𝑥𝑅𝑦} = ∅
10	5, 9	eqtrdi 2782	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∅)
11		0ex 5302	. . . . . . 7 ⊢ ∅ ∈ V
12	10, 11	eqeltrdi 2834	. . . . . 6 ⊢ (𝑧 = ∅ → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
13	12	adantl 480	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 = ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
14		ssv 4003	. . . . . . . . . . . 12 ⊢ 𝑧 ⊆ V
15	14	jctl 522	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
16		vex 3466	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
17	15, 16	jctil 518	. . . . . . . . . 10 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
18		3anass 1092	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ V ∧ (𝑧 ⊆ V ∧ 𝑧 ≠ ∅)))
19	17, 18	sylibr 233	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅))
20		wereu 5668	. . . . . . . . 9 ⊢ ((𝑅 We V ∧ (𝑧 ∈ V ∧ 𝑧 ⊆ V ∧ 𝑧 ≠ ∅)) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
21	19, 20	sylan2 591	. . . . . . . 8 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦)
22		vsnid 4660	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ {𝑤}
23		eleq2 2815	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ↔ 𝑤 ∈ {𝑤}))
24	22, 23	mpbiri 257	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → 𝑤 ∈ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
25		elrabi 3674	. . . . . . . . . . . 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 4916	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = ∪ {𝑤})
28		unisnv 4927	. . . . . . . . . . . 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 510	. . . . . . . . . 10 ⊢ ({𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → (𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
31	30	eximi 1830	. . . . . . . . 9 ⊢ (∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤} → ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
32		reusn 4726	. . . . . . . . 9 ⊢ (∃!𝑦 ∈ 𝑧 ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦 ↔ ∃𝑤{𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = {𝑤})
33		df-rex 3061	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝑧 ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤))
34	31, 32, 33	3imtr4i 291	. . . . . . . 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 2814	. . . . . . . . 9 ⊢ (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤 → (∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
37	36	biimparc 478	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} = 𝑤) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧)
38	37	rexlimiva 3137	. . . . . . 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 3485	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
41	13, 40	pm2.61dane 3019	. . . 4 ⊢ (𝑅 We V → ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
42	41	ralrimivw 3140	. . 3 ⊢ (𝑅 We V → ∀𝑧 ∈ V ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V)
43		wevgblacfn.1	. . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
44	43	fnmpt 6690	. . 3 ⊢ (∀𝑧 ∈ V ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ V → 𝐹 Fn V)
45	42, 44	syl 17	. 2 ⊢ (𝑅 We V → 𝐹 Fn V)
46	43	fvmpt2 7009	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦} ∈ 𝑧) → (𝐹‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
47	16, 39, 46	sylancr 585	. . . . 5 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐹‘𝑧) = ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})
48	47, 39	eqeltrd 2826	. . . 4 ⊢ ((𝑅 We V ∧ 𝑧 ≠ ∅) → (𝐹‘𝑧) ∈ 𝑧)
49	48	ex 411	. . 3 ⊢ (𝑅 We V → (𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
50	49	alrimiv 1923	. 2 ⊢ (𝑅 We V → ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧))
51	45, 50	jca 510	1 ⊢ (𝑅 We V → (𝐹 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1084  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  ∃!wreu 3362  {crab 3419  Vcvv 3462   ⊆ wss 3946  ∅c0 4322  {csn 4623  ∪ cuni 4905   class class class wbr 5143   ↦ cmpt 5226   We wwe 5626   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by: (None)