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Theorem fnunirn 5349
 Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
fnunirn (𝐹 Fn 𝐼 → (A ran 𝐹x 𝐼 A (𝐹x)))
Distinct variable groups:   x,A   x,𝐼   x,𝐹

Proof of Theorem fnunirn
StepHypRef Expression
1 fnfun 4939 . . 3 (𝐹 Fn 𝐼 → Fun 𝐹)
2 elunirn 5348 . . 3 (Fun 𝐹 → (A ran 𝐹x dom 𝐹 A (𝐹x)))
31, 2syl 14 . 2 (𝐹 Fn 𝐼 → (A ran 𝐹x dom 𝐹 A (𝐹x)))
4 fndm 4941 . . 3 (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼)
54rexeqdv 2506 . 2 (𝐹 Fn 𝐼 → (x dom 𝐹 A (𝐹x) ↔ x 𝐼 A (𝐹x)))
63, 5bitrd 177 1 (𝐹 Fn 𝐼 → (A ran 𝐹x 𝐼 A (𝐹x)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  ∃wrex 2301  ∪ cuni 3571  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853 This theorem is referenced by: (None)
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