Theorem ertr 8711

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

Hypothesis

Ref	Expression
ersymb.1	⊢ (𝜑 → 𝑅 Er 𝑋)

Assertion

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

Proof of Theorem ertr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ersymb.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		errel 8705	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
4		simpr 489	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5		brrelex1 5716	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V)
6	3, 4, 5	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐵 ∈ V)
7		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
8		breq2 5114	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
9		breq1 5113	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
10	8, 9	anbi12d 643	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
11	6, 7, 10	spcedv 3558	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))
12		simpl 487	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
13		brrelex1 5716	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
14	3, 12, 13	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴 ∈ V)
15		brrelex2 5717	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
16	3, 4, 15	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐶 ∈ V)
17		brcog 5854	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
18	14, 16, 17	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
19	11, 18	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴(𝑅 ∘ 𝑅)𝐶)
20	19	ex 417	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴(𝑅 ∘ 𝑅)𝐶))
21		df-er 8695	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
22	21	simp3bi 1165	. . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
23	1, 22	syl 18	. . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
24	23	unssbd 4148	. . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
25	24	ssbrd 5155	. 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ 𝑅)𝐶 → 𝐴𝑅𝐶))
26	20, 25	syld 48	1 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906   class class class wbr 5110  ^◡ccnv 5662  dom cdm 5663   ∘ ccom 5667  Rel wrel 5668   Er wer 8692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-co 5672  df-er 8695

This theorem is referenced by:  ertrd  8712  erth  8750  iiner  8788  entr  9004  efginvrel2  19798  efgsrel  19805