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Theorem iserd 8040
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
iserd.1 (𝜑 → Rel 𝑅)
iserd.2 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
iserd.3 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
iserd.4 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
Assertion
Ref Expression
iserd (𝜑𝑅 Er 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iserd
StepHypRef Expression
1 iserd.1 . . 3 (𝜑 → Rel 𝑅)
2 eqidd 2826 . . 3 (𝜑 → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
43ex 403 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
5 iserd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
65ex 403 . . . . . . 7 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
74, 6jca 507 . . . . . 6 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87alrimiv 2026 . . . . 5 (𝜑 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
98alrimiv 2026 . . . 4 (𝜑 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109alrimiv 2026 . . 3 (𝜑 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11 dfer2 8015 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
121, 2, 10, 11syl3anbrc 1447 . 2 (𝜑𝑅 Er dom 𝑅)
1312adantr 474 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 479 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
1513, 14erref 8034 . . . . . . 7 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
1615ex 403 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
17 vex 3417 . . . . . . 7 𝑥 ∈ V
1817, 17breldm 5565 . . . . . 6 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
1916, 18impbid1 217 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
20 iserd.4 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
2119, 20bitr4d 274 . . . 4 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝐴))
2221eqrdv 2823 . . 3 (𝜑 → dom 𝑅 = 𝐴)
23 ereq2 8022 . . 3 (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2422, 23syl 17 . 2 (𝜑 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2512, 24mpbid 224 1 (𝜑𝑅 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1654   = wceq 1656  wcel 2164   class class class wbr 4875  dom cdm 5346  Rel wrel 5351   Er wer 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-er 8014
This theorem is referenced by:  iseri  8041  iseriALT  8042  swoer  8044  iiner  8089  erinxp  8091  cicer  16825  eqger  18002  gaorber  18098  efgrelexlemb  18523  efgcpbllemb  18528  hmpher  21965  xmeter  22615  ercgrg  25836  metider  30478
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