Theorem rnopab 4581

Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
rnopab	⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rnopab

Step	Hyp	Ref	Expression
1		nfopab1 3826	. . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab2 3827	. . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	1, 2	dfrnf 4575	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4		df-br 3765	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabid 3994	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
6	4, 5	bitri 173	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
7	6	exbii 1496	. . 3 ⊢ (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
8	7	abbii 2153	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
9	3, 8	eqtri 2060	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Syntax hints:   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ⟨cop 3378   class class class wbr 3764  {copab 3817  ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  rnmpt  4582  mptpreima  4814  rnoprab  5587