Theorem eqvreltr 38209

Description: An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)

Hypothesis

Ref	Expression
eqvreltr.1	⊢ (𝜑 → EqvRel 𝑅)

Assertion

Ref	Expression
eqvreltr	⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Proof of Theorem eqvreltr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvreltr.1	. . . . . . 7 ⊢ (𝜑 → EqvRel 𝑅)
2		eqvrelrel 38199	. . . . . . 7 ⊢ ( EqvRel 𝑅 → Rel 𝑅)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
4		simpr 483	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5		brrelex1 5731	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V)
6	3, 4, 5	syl2an 594	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐵 ∈ V)
7		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
8		breq2 5153	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
9		breq1 5152	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
10	8, 9	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
11	6, 7, 10	spcedv 3582	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))
12		simpl 481	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
13		brrelex1 5731	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
14	3, 12, 13	syl2an 594	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴 ∈ V)
15		brrelex2 5732	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
16	3, 4, 15	syl2an 594	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐶 ∈ V)
17		brcog 5869	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
18	14, 16, 17	syl2anc 582	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
19	11, 18	mpbird 256	. . 3 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴(𝑅 ∘ 𝑅)𝐶)
20	19	ex 411	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴(𝑅 ∘ 𝑅)𝐶))
21		dfeqvrel2 38192	. . . . . 6 ⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
22	21	simplbi 496	. . . . 5 ⊢ ( EqvRel 𝑅 → (( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅))
23	22	simp3d 1141	. . . 4 ⊢ ( EqvRel 𝑅 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
24	1, 23	syl 17	. . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
25	24	ssbrd 5192	. 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ 𝑅)𝐶 → 𝐴𝑅𝐶))
26	20, 25	syld 47	1 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3461   ⊆ wss 3944   class class class wbr 5149   I cid 5575  ^◡ccnv 5677  dom cdm 5678   ↾ cres 5680   ∘ ccom 5682  Rel wrel 5683   EqvRel weqvrel 37796

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-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

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-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-refrel 38114  df-symrel 38146  df-trrel 38176  df-eqvrel 38187

This theorem is referenced by:  eqvreltrd  38210  eqvrelth  38213