Theorem dmcnvep 38755

Description: Domain of converse epsilon relation. (Contributed by Peter Mazsa, 30-Jan-2018.) (Revised by Peter Mazsa, 23-Nov-2025.)

Assertion

Ref	Expression
dmcnvep	⊢ dom ◡ E = (V ∖ {∅})

Proof of Theorem dmcnvep

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dm 5628	. 2 ⊢ dom ◡ E = {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦}
2		brcnvep 38637	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3436	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	exbii 1855	. . 3 ⊢ (∃𝑦 𝑥◡ E 𝑦 ↔ ∃𝑦 𝑦 ∈ 𝑥)
5	4	abbii 2806	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
6		df-sn 4556	. . . 4 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
7	6	difeq2i 4054	. . 3 ⊢ (V ∖ {∅}) = (V ∖ {𝑥 ∣ 𝑥 = ∅})
8		notab 4242	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = (V ∖ {𝑥 ∣ 𝑥 = ∅})
9		neq0 4280	. . . 4 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
10	9	abbii 2806	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
11	7, 8, 10	3eqtr2ri 2769	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥} = (V ∖ {∅})
12	1, 5, 11	3eqtri 2766	1 ⊢ dom ◡ E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  Vcvv 3431   ∖ cdif 3880  ∅c0 4261  {csn 4555   class class class wbr 5072   E cep 5517  ^◡ccnv 5617  dom cdm 5618

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-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

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-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-eprel 5518  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628

This theorem is referenced by:  dmxrncnvep  38756  dmcnvepres  38757  dmuncnvepres  38758  dfsucmap3  38830