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Theorem rniun 4759
Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rniun ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵

Proof of Theorem rniun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2592 . . . 4 (∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2575 . . . . . 6 𝑧 ∈ V
32elrn2 4601 . . . . 5 (𝑧 ∈ ran 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2346 . . . 4 (∃𝑥𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3686 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1510 . . . 4 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 206 . . 3 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
82elrn2 4601 . . 3 (𝑧 ∈ ran 𝑥𝐴 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3686 . . 3 (𝑧 𝑥𝐴 ran 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
107, 8, 93bitr4i 205 . 2 (𝑧 ∈ ran 𝑥𝐴 𝐵𝑧 𝑥𝐴 ran 𝐵)
1110eqriv 2051 1 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1257  wex 1395  wcel 1407  wrex 2322  cop 3403   ciun 3682  ran crn 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-iun 3684  df-br 3790  df-opab 3844  df-cnv 4378  df-dm 4380  df-rn 4381
This theorem is referenced by:  rnuni  4760  fun11iun  5172
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