Theorem rncnvepres 38904

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 5944	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		cnvepres 38899	. . 3 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
3	2	rneqi 5927	. 2 ⊢ ran (◡ E ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4		dfuni2 4873	. . 3 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5		df-rex 3088	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
6	5	abbii 2828	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
7	4, 6	eqtri 2784	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
8	1, 3, 7	3eqtr4i 2794	1 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  {cab 2739  ∃wrex 3087  ∪ cuni 4871  {copab 5172   E cep 5560  ^◡ccnv 5660  ran crn 5662   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673

This theorem is referenced by:  dm1cosscnvepres  39141