Theorem iseri 8668

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8667, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
iseri.1	⊢ Rel 𝑅
iseri.2	⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)
iseri.3	⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
iseri.4	⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)

Assertion

Ref	Expression
iseri	⊢ 𝑅 Er 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴

Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iseri

Step	Hyp	Ref	Expression
1		iseri.1	. . . 4 ⊢ Rel 𝑅
2	1	a1i 11	. . 3 ⊢ (⊤ → Rel 𝑅)
3		iseri.2	. . . 4 ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥)
4	3	adantl 482	. . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5		iseri.3	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
6	5	adantl 482	. . 3 ⊢ ((⊤ ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7		iseri.4	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)
8	7	a1i 11	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥))
9	2, 4, 6, 8	iserd 8667	. 2 ⊢ (⊤ → 𝑅 Er 𝐴)
10	9	mptru 1554	1 ⊢ 𝑅 Er 𝐴

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ⊤wtru 1548   ∈ wcel 2119   class class class wbr 5079  Rel wrel 5630   Er wer 8637

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-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-er 8640

This theorem is referenced by:  brinxper  8670  eqer  8677  0er  8679  ecopover  8765  ener  8945  gicer  19250  hmpher  23774  phtpcer  24987  vitalilem1  25600  tgjustf  28566  erclwwlk  30118  erclwwlkn  30167