Theorem dmcnvep 39037

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 5671	. 2 ⊢ dom ◡ E = {𝑥 ∣ ∃𝑦 𝑥◡ E 𝑦}
2		brcnvep 38919	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥))
3	2	elv 3460	. . . 4 ⊢ (𝑥◡ E 𝑦 ↔ 𝑦 ∈ 𝑥)
4	3	exbii 1878	. . 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 4590	. . . 4 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
7	6	difeq2i 4078	. . 3 ⊢ (V ∖ {∅}) = (V ∖ {𝑥 ∣ 𝑥 = ∅})
8		notab 4267	. . 3 ⊢ {𝑥 ∣ ¬ 𝑥 = ∅} = (V ∖ {𝑥 ∣ 𝑥 = ∅})
9		neq0 4306	. . . 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 ∖ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  {csn 4589   class class class wbr 5109   E cep 5560  ^◡ccnv 5660  dom cdm 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671

This theorem is referenced by:  dmxrncnvep  39038  dmcnvepres  39039  dmuncnvepres  39040  dfsucmap3  39112