Theorem ordtbas2 23181

Description: Lemma for ordtbas 23182. (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 4114	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
2		ssun2 4115	. . . . . . 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 23180	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))
7		dmexg 7848	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
8	3, 7	eqeltrid 2844	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V)
9	6, 8	eqeltrrd 2841	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
10		uniexb 7714	. . . . . . . 8 ⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
11	9, 10	sylibr 235	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V)
12		ssexg 5258	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
13	2, 11, 12	sylancr 593	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V)
14		ssexg 5258	. . . . . 6 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
15	1, 13, 14	sylancr 593	. . . . 5 ⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V)
16		ssun2 4115	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
17		ssexg 5258	. . . . . 6 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
18	16, 13, 17	sylancr 593	. . . . 5 ⊢ (𝑅 ∈ TosetRel → 𝐵 ∈ V)
19		elfiun 9340	. . . . 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 590	. . . 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 23178	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = 𝐴)
22	21, 1	eqsstrdi 3966	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ (𝐴 ∪ 𝐵))
23		ssun1 4114	. . . . . . 7 ⊢ (𝐴 ∪ 𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)
24	22, 23	sstrdi 3934	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
25	24	sseld 3921	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
26		cnvtsr 18552	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
27		df-rn 5636	. . . . . . . . . . 11 ⊢ ran 𝑅 = dom ◡𝑅
28		eqid 2740	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
29	27, 28	ordtbaslem 23178	. . . . . . . . . 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 18551	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
32	3	psrn 18539	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅)
34		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
35		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
36	34, 35	brcnv 5831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
37	36	bicomi 225	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥)
38	37	notbii 321	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥))
40	33, 39	rabeqbidv 3410	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})
41	33, 40	mpteq12dv 5166	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
42	41	rneqd 5887	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
43	5, 42	eqtrid 2787	. . . . . . . . . 10 ⊢ (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))
44	43	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})))
45	30, 44, 43	3eqtr4d 2785	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = 𝐵)
46	45, 16	eqsstrdi 3966	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ (𝐴 ∪ 𝐵))
47	46, 23	sstrdi 3934	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
48	47	sseld 3921	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
49		ssun2 4115	. . . . . . . 8 ⊢ 𝐶 ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)
50	21, 4	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐴) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
51	50	eleq2d 2826	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})))
52		breq2 5083	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎))
53	52	notbid 319	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎))
54	53	rabbidv 3399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
55	54	cbvmptv 5183	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
56	55	elrnmpt 5907	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ V → (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}))
57	56	elv 3437	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
58	51, 57	bitrdi 288	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}))
59	45, 5	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ TosetRel → (fi‘𝐵) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
60	59	eleq2d 2826	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})))
61		breq1 5082	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦))
62	61	notbid 319	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦))
63	62	rabbidv 3399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
64	63	cbvmptv 5183	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
65	64	elrnmpt 5907	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
66	65	elv 3437	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
67	60, 66	bitrdi 288	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
68	58, 67	anbi12d 638	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})))
69		reeanv 3212	. . . . . . . . . . . . 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 4151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))
71		inrab 4251	. . . . . . . . . . . . . . . 16 ⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}
72	70, 71	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
73	72	reximi 3078	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
74	73	reximi 3078	. . . . . . . . . . . . 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 236	. . . . . . . . . . . 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	biimtrdi 254	. . . . . . . . . . 11 ⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}))
77	76	imp 407	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
78		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
79	78	inex1 5252	. . . . . . . . . . 11 ⊢ (𝑚 ∩ 𝑛) ∈ V
80		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
81	80	elrnmpog 7498	. . . . . . . . . . 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 235	. . . . . . . . 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 2851	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ 𝐶)
86	49, 85	sselid 3920	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))
87		eleq1 2828	. . . . . . 7 ⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶) ↔ (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
88	86, 87	syl5ibrcom 248	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
89	88	rexlimdvva 3197	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
90	25, 48, 89	3jaod 1437	. . . 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 241	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)))
92	91	ssrdv 3928	. 2 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶))
93		ssfii 9329	. . . 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 481	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵)))
96		simprl 776	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋)
97		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})
98	54	rspceeqv 3590	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
99	96, 97, 98	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
100	8	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V)
101		rabexg 5272	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V)
102		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
103	102	elrnmpt 5907	. . . . . . . . . . . . . 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 258	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))
106	105, 4	eleqtrrdi 2851	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴)
107	1, 106	sselid 3920	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴 ∪ 𝐵))
108	95, 107	sseldd 3923	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)))
109		simprr 778	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋)
110		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})
111	63	rspceeqv 3590	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
112	109, 110, 111	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
113		rabexg 5272	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V)
114		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
115	114	elrnmpt 5907	. . . . . . . . . . . . . 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 258	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))
118	117, 5	eleqtrrdi 2851	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵)
119	16, 118	sselid 3920	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴 ∪ 𝐵))
120	95, 119	sseldd 3923	. . . . . . . . 9 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵)))
121		fiin 9332	. . . . . . . . 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 590	. . . . . . . 8 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵)))
123	71, 122	eqeltrrid 2845	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)))
124	123	ralrimivva 3183	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)))
125	80	fmpo 8017	. . . . . 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 219	. . . . 5 ⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵)))
127	126	frnd 6670	. . . 4 ⊢ (𝑅 ∈ TosetRel → ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵)))
128	84, 127	eqsstrid 3960	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴 ∪ 𝐵)))
129	94, 128	unssd 4128	. 2 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴 ∪ 𝐵)))
130	92, 129	eqssd 3939	1 ⊢ (𝑅 ∈ TosetRel → (fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1091   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  {crab 3392  Vcvv 3432   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  {csn 4562  ∪ cuni 4845   class class class wbr 5079   ↦ cmpt 5160   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  ⟶wf 6488  ‘cfv 6492   ∈ cmpo 7365  ficfi 9320  PosetRelcps 18528   TosetRel ctsr 18529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-1o 8402  df-2o 8403  df-en 8891  df-fin 8894  df-fi 9321  df-ps 18530  df-tsr 18531

This theorem is referenced by:  ordtbas  23182  leordtval  23203