Theorem ordtbas2 21793

Description: Lemma for ordtbas 21794. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
ordtval.1	⊢ 𝑋 = dom 𝑅
ordtval.2	⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3	⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
ordtval.4	⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})

Assertion

Ref	Expression
ordtbas2	⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))

Distinct variable groups:   𝑎,𝑏,𝐴   𝑥,𝑎,𝑦,𝑅,𝑏   𝑋,𝑎,𝑏,𝑥,𝑦   𝐵,𝑎,𝑏

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ordtbas2

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 4148	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		ssun2 4149	. . . . . . 7 ⊢ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵))
3		ordtval.1	. . . . . . . . . 10 ⊢ 𝑋 = dom 𝑅
4		ordtval.2	. . . . . . . . . 10 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
5		ordtval.3	. . . . . . . . . 10 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
6	3, 4, 5	ordtuni 21792	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7607	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2917	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2914	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7480	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 236	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5220	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 589	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		ssexg 5220	. . . . . 6 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
15	1, 13, 14	sylancr 589	. . . . 5 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16		ssun2 4149	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
17		ssexg 5220	. . . . . 6 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
18	16, 13, 17	sylancr 589	. . . . 5 ⊢ (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19		elfiun 8888	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛))))
20	15, 18, 19	syl2anc 586	. . . 4 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛))))
21	3, 4	ordtbaslem 21790	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
22	21, 1	eqsstrdi 4021	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴 ∪ 𝐵))
23		ssun1 4148	. . . . . . 7 ⊢ (𝐴 ∪ 𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)
24	22, 23	sstrdi 3979	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
25	24	sseld 3966	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
26		cnvtsr 17826	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
27		df-rn 5561	. . . . . . . . . . 11 ⊢ ran 𝑅 = dom ◡𝑅
28		eqid 2821	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
29	27, 28	ordtbaslem 21790	. . . . . . . . . 10 ⊢ (◡𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
30	26, 29	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
31		tsrps 17825	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
32	3	psrn 17813	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34		vex 3498	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
35		vex 3498	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
36	34, 35	brcnv 5748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
37	36	bicomi 226	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥)
38	37	notbii 322	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥))
40	33, 39	rabeqbidv 3486	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
41	33, 40	mpteq12dv 5144	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
42	41	rneqd 5803	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
43	5, 42	syl5eq 2868	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
44	43	fveq2d 6669	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})))
45	30, 44, 43	3eqtr4d 2866	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
46	45, 16	eqsstrdi 4021	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴 ∪ 𝐵))
47	46, 23	sstrdi 3979	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
48	47	sseld 3966	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
49		ssun2 4149	. . . . . . . 8 ⊢ 𝐶 ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)
50	21, 4	syl6eq 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
51	50	eleq2d 2898	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52		breq2 5063	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
53	52	notbid 320	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
54	53	rabbidv 3481	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
55	54	cbvmptv 5162	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
56	55	elrnmpt 5823	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ V → (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}))
57	56	elv 3500	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
58	51, 57	syl6bb 289	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}))
59	45, 5	syl6eq 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
60	59	eleq2d 2898	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))
61		breq1 5062	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦))
62	61	notbid 320	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
63	62	rabbidv 3481	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
64	63	cbvmptv 5162	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
65	64	elrnmpt 5823	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
66	65	elv 3500	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
67	60, 66	syl6bb 289	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
68	58, 67	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})))
69		reeanv 3368	. . . . . . . . . . . . 13 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
70		ineq12 4184	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
71		inrab 4275	. . . . . . . . . . . . . . . 16 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
72	70, 71	syl6eq 2872	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
73	72	reximi 3243	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
74	73	reximi 3243	. . . . . . . . . . . . 13 ⊢ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
75	69, 74	sylbir 237	. . . . . . . . . . . 12 ⊢ ((∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
76	68, 75	syl6bi 255	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
77	76	imp 409	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
78		vex 3498	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
79	78	inex1 5214	. . . . . . . . . . 11 ⊢ (𝑚 ∩ 𝑛) ∈ V
80		eqid 2821	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
81	80	elrnmpog 7280	. . . . . . . . . . 11 ⊢ ((𝑚 ∩ 𝑛) ∈ V → ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
82	79, 81	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
83	77, 82	sylibr 236	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
84		ordtval.4	. . . . . . . . 9 ⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
85	83, 84	eleqtrrdi 2924	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ 𝐶)
86	49, 85	sseldi 3965	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))
87		eleq1 2900	. . . . . . 7 ⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶) ↔ (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
88	86, 87	syl5ibrcom 249	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
89	88	rexlimdvva 3294	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
90	25, 48, 89	3jaod 1424	. . . 4 ⊢ (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
91	20, 90	sylbid 242	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
92	91	ssrdv 3973	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
93		ssfii 8877	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵)))
94	13, 93	syl 17	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵)))
95	94	adantr 483	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵)))
96		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
97		eqidd 2822	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
98	54	rspceeqv 3638	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
99	96, 97, 98	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
100	8	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
101		rabexg 5227	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
102		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
103	102	elrnmpt 5823	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
104	100, 101, 103	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
105	99, 104	mpbird 259	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106	105, 4	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
107	1, 106	sseldi 3965	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴 ∪ 𝐵))
108	95, 107	sseldd 3968	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)))
109		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
110		eqidd 2822	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
111	63	rspceeqv 3638	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
112	109, 110, 111	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
113		rabexg 5227	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
114		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
115	114	elrnmpt 5823	. . . . . . . . . . . . . 14 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
116	100, 113, 115	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
117	112, 116	mpbird 259	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
118	117, 5	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
119	16, 118	sseldi 3965	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴 ∪ 𝐵))
120	95, 119	sseldd 3968	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵)))
121		fiin 8880	. . . . . . . . 9 ⊢ (({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)) ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵)))
122	108, 120, 121	syl2anc 586	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵)))
123	71, 122	eqeltrrid 2918	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)))
124	123	ralrimivva 3191	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)))
125	80	fmpo 7760	. . . . . 6 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵)))
126	124, 125	sylib 220	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵)))
127	126	frnd 6516	. . . 4 ⊢ (𝑅 ∈ TosetRel → ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵)))
128	84, 127	eqsstrid 4015	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴 ∪ 𝐵)))
129	94, 128	unssd 4162	. 2 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴 ∪ 𝐵)))
130	92, 129	eqssd 3984	1 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ w3o 1082   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3495   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  {csn 4561  ∪ cuni 4832   class class class wbr 5059   ↦ cmpt 5139   × cxp 5548  ^◡ccnv 5549  dom cdm 5550  ran crn 5551  ⟶wf 6346  ‘cfv 6350   ∈ cmpo 7152  ficfi 8868  PosetRelcps 17802   TosetRel ctsr 17803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-fin 8507  df-fi 8869  df-ps 17804  df-tsr 17805

This theorem is referenced by:  ordtbas  21794  leordtval  21815