Theorem joinfval 17469

Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 17470 first to reduce net proof size (existence part)?

Hypotheses

Ref	Expression
joinfval.u	⊢ 𝑈 = (lub‘𝐾)
joinfval.j	⊢ ∨ = (join‘𝐾)

Assertion

Ref	Expression
joinfval	⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})

Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑧,𝑈

Allowed substitution hints:   𝑈(𝑥,𝑦)   ∨ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem joinfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3433	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		joinfval.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		fvex 6512	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
4		moeq 3612	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))
6		eqid 2778	. . . . . . 7 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))}
7	3, 3, 5, 6	oprabex 7489	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))} ∈ V)
9		joinfval.u	. . . . . . . . . . . 12 ⊢ 𝑈 = (lub‘𝐾)
10	9	lubfun 17448	. . . . . . . . . . 11 ⊢ Fun 𝑈
11		funbrfv2b 6553	. . . . . . . . . . 11 ⊢ (Fun 𝑈 → ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧)))
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ ({𝑥, 𝑦}𝑈𝑧 ↔ ({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧))
13		eqid 2778	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
14		eqid 2778	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) = (le‘𝐾)
15		simpl 475	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → 𝐾 ∈ V)
16		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ∈ dom 𝑈)
17	13, 14, 9, 15, 16	lubelss 17450	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
18	17	ex 405	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19		vex 3418	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		vex 3418	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	19, 20	prss 4627	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
22	18, 21	syl6ibr 244	. . . . . . . . . . 11 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23		eqcom 2785	. . . . . . . . . . . 12 ⊢ ((𝑈‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝑈‘{𝑥, 𝑦}))
24	23	biimpi 208	. . . . . . . . . . 11 ⊢ ((𝑈‘{𝑥, 𝑦}) = 𝑧 → 𝑧 = (𝑈‘{𝑥, 𝑦}))
25	22, 24	anim12d1 600	. . . . . . . . . 10 ⊢ (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
26	12, 25	syl5bi 234	. . . . . . . . 9 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
27	26	alrimiv 1886	. . . . . . . 8 ⊢ (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
28	27	alrimiv 1886	. . . . . . 7 ⊢ (𝐾 ∈ V → ∀𝑦∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
29	28	alrimiv 1886	. . . . . 6 ⊢ (𝐾 ∈ V → ∀𝑥∀𝑦∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
30		ssoprab2 7041	. . . . . 6 ⊢ (∀𝑥∀𝑦∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
31	29, 30	syl 17	. . . . 5 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))})
32	8, 31	ssexd 5084	. . . 4 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V)
33		fveq2 6499	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
34	33, 9	syl6eqr 2832	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
35	34	breqd 4940	. . . . . 6 ⊢ (𝑝 = 𝐾 → ({𝑥, 𝑦} (lub‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝑈𝑧))
36	35	oprabbidv 7039	. . . . 5 ⊢ (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
37		df-join 17444	. . . . 5 ⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
38	36, 37	fvmptg 6593	. . . 4 ⊢ ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V) → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
39	32, 38	mpdan 674	. . 3 ⊢ (𝐾 ∈ V → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
40	2, 39	syl5eq 2826	. 2 ⊢ (𝐾 ∈ V → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
41	1, 40	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → ∨ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507   ∈ wcel 2050  ∃*wmo 2545  Vcvv 3415   ⊆ wss 3829  {cpr 4443   class class class wbr 4929  dom cdm 5407  Fun wfun 6182  ‘cfv 6188  {coprab 6977  Basecbs 16339  lecple 16428  lubclub 17410  joincjn 17412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-oprab 6980  df-lub 17442  df-join 17444

This theorem is referenced by:  joinfval2  17470  join0  17606  odumeet  17608  odujoin  17610