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Theorem fnasrng 5343
Description: A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.)
Assertion
Ref Expression
fnasrng (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩))

Proof of Theorem fnasrng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfmptg 5342 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩})
2 eqid 2040 . . . . 5 (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩)
32rnmpt 4582 . . . 4 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
4 velsn 3392 . . . . . 6 (𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ 𝑦 = ⟨𝑥, 𝐵⟩)
54rexbii 2331 . . . . 5 (∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩} ↔ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩)
65abbii 2153 . . . 4 {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = ⟨𝑥, 𝐵⟩}
73, 6eqtr4i 2063 . . 3 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
8 df-iun 3659 . . 3 𝑥𝐴 {⟨𝑥, 𝐵⟩} = {𝑦 ∣ ∃𝑥𝐴 𝑦 ∈ {⟨𝑥, 𝐵⟩}}
97, 8eqtr4i 2063 . 2 ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
101, 9syl6eqr 2090 1 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) = ran (𝑥𝐴 ↦ ⟨𝑥, 𝐵⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  {csn 3375  cop 3378   ciun 3657  cmpt 3818  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-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  resfunexg  5382
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