Theorem rncnvepres 37662

Description: The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
rncnvepres	⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴

Proof of Theorem rncnvepres

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

Step	Hyp	Ref	Expression
1		rnopab 5943	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		cnvepres 37657	. . 3 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
3	2	rneqi 5926	. 2 ⊢ ran (◡ E ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4		dfuni2 4901	. . 3 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5		df-rex 3063	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
6	5	abbii 2794	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
7	4, 6	eqtri 2752	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
8	1, 3, 7	3eqtr4i 2762	1 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701  ∃wrex 3062  ∪ cuni 4899  {copab 5200   E cep 5569  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-eprel 5570  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678

This theorem is referenced by:  dm1cosscnvepres  37816