Theorem iseri 8483

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8482, 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 481	. . 3 ⊢ ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5		iseri.3	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
6	5	adantl 481	. . 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 8482	. 2 ⊢ (⊤ → 𝑅 Er 𝐴)
10	9	mptru 1546	1 ⊢ 𝑅 Er 𝐴

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ⊤wtru 1540   ∈ wcel 2108   class class class wbr 5070  Rel wrel 5585   Er wer 8453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-er 8456

This theorem is referenced by:  eqer  8491  0er  8493  ecopover  8568  ener  8742  gicer  18807  hmpher  22843  phtpcer  24064  vitalilem1  24677  tgjustf  26738  erclwwlk  28288  erclwwlkn  28337