Theorem joinfval 18386

Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 18387 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 3474	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		joinfval.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		fvex 6876	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
4		moeq 3669	. . . . . . . 8 ⊢ ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦})
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → ∃*𝑧 𝑧 = (𝑈‘{𝑥, 𝑦}))
6		eqid 2761	. . . . . . 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 7953	. . . . . 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 18365	. . . . . . . . . . 11 ⊢ Fun 𝑈
11		funbrfv2b 6920	. . . . . . . . . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
14		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) = (le‘𝐾)
15		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → 𝐾 ∈ V)
16		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ∈ dom 𝑈)
17	13, 14, 9, 15, 16	lubelss 18367	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ V ∧ {𝑥, 𝑦} ∈ dom 𝑈) → {𝑥, 𝑦} ⊆ (Base‘𝐾))
18	17	ex 416	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → {𝑥, 𝑦} ⊆ (Base‘𝐾)))
19		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	19, 20	prss 4777	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ↔ {𝑥, 𝑦} ⊆ (Base‘𝐾))
22	18, 21	imbitrrdi 254	. . . . . . . . . . 11 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦} ∈ dom 𝑈 → (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))))
23		eqcom 2768	. . . . . . . . . . . 12 ⊢ ((𝑈‘{𝑥, 𝑦}) = 𝑧 ↔ 𝑧 = (𝑈‘{𝑥, 𝑦}))
24	23	biimpi 218	. . . . . . . . . . 11 ⊢ ((𝑈‘{𝑥, 𝑦}) = 𝑧 → 𝑧 = (𝑈‘{𝑥, 𝑦}))
25	22, 24	anim12d1 619	. . . . . . . . . 10 ⊢ (𝐾 ∈ V → (({𝑥, 𝑦} ∈ dom 𝑈 ∧ (𝑈‘{𝑥, 𝑦}) = 𝑧) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
26	12, 25	biimtrid 244	. . . . . . . . 9 ⊢ (𝐾 ∈ V → ({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
27	26	alrimiv 1946	. . . . . . . 8 ⊢ (𝐾 ∈ V → ∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
28	27	alrimiv 1946	. . . . . . 7 ⊢ (𝐾 ∈ V → ∀𝑦∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
29	28	alrimiv 1946	. . . . . 6 ⊢ (𝐾 ∈ V → ∀𝑥∀𝑦∀𝑧({𝑥, 𝑦}𝑈𝑧 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 = (𝑈‘{𝑥, 𝑦}))))
30		ssoprab2 7460	. . . . . 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 5279	. . . 4 ⊢ (𝐾 ∈ V → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V)
33		fveq2 6863	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = (lub‘𝐾))
34	33, 9	eqtr4di 2814	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lub‘𝑝) = 𝑈)
35	34	breqd 5110	. . . . . 6 ⊢ (𝑝 = 𝐾 → ({𝑥, 𝑦} (lub‘𝑝)𝑧 ↔ {𝑥, 𝑦}𝑈𝑧))
36	35	oprabbidv 7458	. . . . 5 ⊢ (𝑝 = 𝐾 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
37		df-join 18361	. . . . 5 ⊢ join = (𝑝 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧})
38	36, 37	fvmptg 6969	. . . 4 ⊢ ((𝐾 ∈ V ∧ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧} ∈ V) → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
39	32, 38	mpdan 697	. . 3 ⊢ (𝐾 ∈ V → (join‘𝐾) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦}𝑈𝑧})
40	2, 39	eqtrid 2808	. 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 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∃*wmo 2563  Vcvv 3453   ⊆ wss 3904  {cpr 4583   class class class wbr 5099  dom cdm 5645  Fun wfun 6511  ‘cfv 6517  {coprab 7393  Basecbs 17228  lecple 17276  lubclub 18324  joincjn 18326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-oprab 7396  df-lub 18359  df-join 18361

This theorem is referenced by:  joinfval2  18387  join0  18418  odujoin  18421  odumeet  18423