Theorem dmcnvep 38639

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 5642	. 2 ⊢ dom ◡ E = {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦}
2		brcnvep 38521	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3447	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	exbii 1850	. . 3 ⊢ (∃𝑦 𝑥◡ E 𝑦 ↔ ∃𝑦 𝑦 ∈ 𝑥)
5	4	abbii 2804	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
6		df-sn 4583	. . . 4 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
7	6	difeq2i 4077	. . 3 ⊢ (V ∖ {∅}) = (V ∖ {𝑥 ∣ 𝑥 = ∅})
8		notab 4268	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = (V ∖ {𝑥 ∣ 𝑥 = ∅})
9		neq0 4306	. . . 4 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
10	9	abbii 2804	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥}
11	7, 8, 10	3eqtr2ri 2767	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑦 ∈ 𝑥} = (V ∖ {∅})
12	1, 5, 11	3eqtri 2764	1 ⊢ dom ◡ E = (V ∖ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  Vcvv 3442   ∖ cdif 3900  ∅c0 4287  {csn 4582   class class class wbr 5100   E cep 5531  ^◡ccnv 5631  dom cdm 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642

This theorem is referenced by:  dmxrncnvep  38640  dmcnvepres  38641  dmuncnvepres  38642  dfsucmap3  38714