Theorem dmcnvep 38356

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 5650	. 2 ⊢ dom ◡ E = {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦}
2		brcnvep 38249	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3455	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	exbii 1848	. . 3 ⊢ (∃𝑦 𝑥◡ E 𝑦 ↔ ∃𝑦 𝑦 ∈ 𝑥)
5	4	abbii 2797	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
6		df-sn 4592	. . . 4 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
7	6	difeq2i 4088	. . 3 ⊢ (V ∖ {∅}) = (V ∖ {𝑥 ∣ 𝑥 = ∅})
8		notab 4279	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = (V ∖ {𝑥 ∣ 𝑥 = ∅})
9		neq0 4317	. . . 4 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
10	9	abbii 2797	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
11	7, 8, 10	3eqtr2ri 2760	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥} = (V ∖ {∅})
12	1, 5, 11	3eqtri 2757	1 ⊢ dom ◡ E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  Vcvv 3450   ∖ cdif 3913  ∅c0 4298  {csn 4591   class class class wbr 5109   E cep 5539  ^◡ccnv 5639  dom cdm 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-eprel 5540  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650

This theorem is referenced by:  dmxrncnvep  38357