Theorem iseri 8731

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8730, 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 487	. . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5		iseri.3	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
6	5	adantl 487	. . 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 8730	. 2 ⊢ (⊤ → 𝑅 Er 𝐴)
10	9	mptru 1577	1 ⊢ 𝑅 Er 𝐴

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ⊤wtru 1571   ∈ wcel 2146   class class class wbr 5114  Rel wrel 5671   Er wer 8700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-er 8703

This theorem is used by:  brinxper  8733  eqer  8740  0er  8742  ecopover  8828  ener  9007  gicer  19378  ricer  20641  hmpher  23978  phtpcer  25191  vitalilem1  25804  tgjustf  28779  erclwwlk  30411  erclwwlkn  30460