Theorem iseri 8701

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8700, 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 485	. . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5		iseri.3	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
6	5	adantl 485	. . 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 8700	. 2 ⊢ (⊤ → 𝑅 Er 𝐴)
10	9	mptru 1566	1 ⊢ 𝑅 Er 𝐴

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ⊤wtru 1560   ∈ wcel 2141   class class class wbr 5099  Rel wrel 5650   Er wer 8670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-er 8673

This theorem is referenced by:  brinxper  8703  eqer  8710  0er  8712  ecopover  8798  ener  8978  gicer  19300  hmpher  23824  phtpcer  25037  vitalilem1  25650  tgjustf  28619  erclwwlk  30171  erclwwlkn  30220