Theorem rnxrncnvepres 38467

Description: Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.)

Assertion

Ref	Expression
rnxrncnvepres	⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)}

Distinct variable groups:   𝑢,𝐴,𝑥,𝑦   𝑢,𝑅,𝑥,𝑦

Proof of Theorem rnxrncnvepres

Step	Hyp	Ref	Expression
1		rnxrnres 38466	. 2 ⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦)}
2		brcnvep 38322	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢))
3	2	elv 3442	. . . . 5 ⊢ (𝑢◡ E 𝑦 ↔ 𝑦 ∈ 𝑢)
4	3	anbi1ci 626	. . . 4 ⊢ ((𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦) ↔ (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥))
5	4	rexbii 3080	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦) ↔ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥))
6	5	opabbii 5160	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢◡ E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)}
7	1, 6	eqtri 2756	1 ⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437   class class class wbr 5093  {copab 5155   E cep 5518  ^◡ccnv 5618  ran crn 5620   ↾ cres 5621   ⋉ cxrn 38234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-1st 7927  df-2nd 7928  df-ec 8630  df-xrn 38424

This theorem is referenced by: (None)