Theorem eqvrelth 38111

Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)

Hypotheses

Ref	Expression
eqvrelth.1	⊢ (𝜑 → EqvRel 𝑅)
eqvrelth.2	⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Assertion

Ref	Expression
eqvrelth	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem eqvrelth

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvrelth.1	. . . . . . . 8 ⊢ (𝜑 → EqvRel 𝑅)
2	1	eqvrelsymb 38106	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
3	2	biimpa 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵𝑅𝐴)
4	1	eqvreltr 38107	. . . . . . 7 ⊢ (𝜑 → ((𝐵𝑅𝐴 ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥))
5	4	impl 454	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵𝑅𝐴) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
6	3, 5	syldanl 600	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐴𝑅𝑥) → 𝐵𝑅𝑥)
7	1	eqvreltr 38107	. . . . . 6 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥))
8	7	impl 454	. . . . 5 ⊢ (((𝜑 ∧ 𝐴𝑅𝐵) ∧ 𝐵𝑅𝑥) → 𝐴𝑅𝑥)
9	6, 8	impbida 799	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝐴𝑅𝑥 ↔ 𝐵𝑅𝑥))
10		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
11		eqvrelth.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
12	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
13		elecg 8764	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ dom 𝑅) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
14	10, 12, 13	sylancr 585	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
15		eqvrelrel 38097	. . . . . . 7 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
17		brrelex2 5724	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
18	16, 17	sylan 578	. . . . 5 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
19		elecg 8764	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ V) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
20	10, 18, 19	sylancr 585	. . . 4 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑥))
21	9, 14, 20	3bitr4d 310	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝑥 ∈ [𝐵]𝑅))
22	21	eqrdv 2723	. 2 ⊢ ((𝜑 ∧ 𝐴𝑅𝐵) → [𝐴]𝑅 = [𝐵]𝑅)
23	1	adantr 479	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → EqvRel 𝑅)
24	1, 11	eqvrelref 38110	. . . . . . 7 ⊢ (𝜑 → 𝐴𝑅𝐴)
25	24	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐴)
26	11	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ dom 𝑅)
27		elecALTV 37766	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝑅 ∧ 𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
28	26, 26, 27	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐴))
29	25, 28	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐴]𝑅)
30		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → [𝐴]𝑅 = [𝐵]𝑅)
31	29, 30	eleqtrd 2827	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴 ∈ [𝐵]𝑅)
32	30	dmec2d 37805	. . . . . 6 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ dom 𝑅 ↔ 𝐵 ∈ dom 𝑅))
33	26, 32	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵 ∈ dom 𝑅)
34		elecALTV 37766	. . . . 5 ⊢ ((𝐵 ∈ dom 𝑅 ∧ 𝐴 ∈ dom 𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
35	33, 26, 34	syl2anc 582	. . . 4 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
36	31, 35	mpbid 231	. . 3 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐵𝑅𝐴)
37	23, 36	eqvrelsym 38105	. 2 ⊢ ((𝜑 ∧ [𝐴]𝑅 = [𝐵]𝑅) → 𝐴𝑅𝐵)
38	22, 37	impbida 799	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   class class class wbr 5141  dom cdm 5670  Rel wrel 5675  [cec 8719   EqvRel weqvrel 37694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ec 8723  df-refrel 38012  df-symrel 38044  df-trrel 38074  df-eqvrel 38085

This theorem is referenced by:  eqvrelthi  38113