Theorem ertr 6708

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 6702	. . . . . . 7 ⊢ (𝑅 Er 𝑋 → Rel 𝑅)
3	1, 2	syl 14	. . . . . 6 ⊢ (𝜑 → Rel 𝑅)
4		simpr 110	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
5		brrelex 4761	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V)
6	3, 4, 5	syl2an 289	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐵 ∈ V)
7		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶))
8		breq2 4087	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
9		breq1 4086	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐶 ↔ 𝐵𝑅𝐶))
10	8, 9	anbi12d 473	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
11	10	spcegv 2891	. . . . 5 ⊢ (𝐵 ∈ V → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
12	6, 7, 11	sylc 62	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶))
13		simpl 109	. . . . . 6 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
14		brrelex 4761	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
15	3, 13, 14	syl2an 289	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴 ∈ V)
16		brrelex2 4762	. . . . . 6 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐶 ∈ V)
17	3, 4, 16	syl2an 289	. . . . 5 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐶 ∈ V)
18		brcog 4892	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
19	15, 17, 18	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → (𝐴(𝑅 ∘ 𝑅)𝐶 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥𝑅𝐶)))
20	12, 19	mpbird 167	. . 3 ⊢ ((𝜑 ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴(𝑅 ∘ 𝑅)𝐶)
21	20	ex 115	. 2 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴(𝑅 ∘ 𝑅)𝐶))
22		df-er 6693	. . . . . 6 ⊢ (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
23	22	simp3bi 1038	. . . . 5 ⊢ (𝑅 Er 𝑋 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
24	1, 23	syl 14	. . . 4 ⊢ (𝜑 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)
25	24	unssbd 3382	. . 3 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
26	25	ssbrd 4126	. 2 ⊢ (𝜑 → (𝐴(𝑅 ∘ 𝑅)𝐶 → 𝐴𝑅𝐶))
27	21, 26	syld 45	1 ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197   class class class wbr 4083  ^◡ccnv 4719  dom cdm 4720   ∘ ccom 4724  Rel wrel 4725   Er wer 6690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4726  df-rel 4727  df-co 4729  df-er 6693

This theorem is referenced by:  ertrd  6709  erth  6739  iinerm  6767  entr  6949