Theorem rncnvepres 38284

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 5907	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
2		cnvepres 38279	. . 3 ⊢ (◡ E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
3	2	rneqi 5890	. 2 ⊢ ran (◡ E ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)}
4		dfuni2 4869	. . 3 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
5		df-rex 3054	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
6	5	abbii 2796	. . 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 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∃wrex 3053  ∪ cuni 4867  {copab 5164   E cep 5530  ^◡ccnv 5630  ran crn 5632   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643

This theorem is referenced by:  dm1cosscnvepres  38440