Theorem dmcnvep 39586

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 5668	. 2 ⊢ dom ◡ E = {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦}
2		brcnvep 39468	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3459	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	exbii 1881	. . 3 ⊢ (∃𝑦 𝑥◡ E 𝑦 ↔ ∃𝑦 𝑦 ∈ 𝑥)
5	4	abbii 2830	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
6		df-sn 4589	. . . 4 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
7	6	difeq2i 4063	. . 3 ⊢ (V ∖ {∅}) = (V ∖ {𝑥 ∣ 𝑥 = ∅})
8		notab 4262	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = (V ∖ {𝑥 ∣ 𝑥 = ∅})
9		neq0 4302	. . . 4 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
10	9	abbii 2830	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
11	7, 8, 10	3eqtr2ri 2793	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥} = (V ∖ {∅})
12	1, 5, 11	3eqtri 2790	1 ⊢ dom ◡ E = (V ∖ {∅})

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   ∖ cdif 4048  ∅c0 4281  {csn 4588   class class class wbr 5108   E cep 5557  ^◡ccnv 5657  dom cdm 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668

This theorem is used by:  dmxrncnvep  39587  dmcnvepres  39588  dmuncnvepres  39589  dfsucmap3  39661