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Theorem fvexg 5137
Description: Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
Assertion
Ref Expression
fvexg ((𝐹 𝑉 A 𝑊) → (𝐹A) V)

Proof of Theorem fvexg
StepHypRef Expression
1 elex 2560 . . 3 (A 𝑊A V)
2 fvssunirng 5133 . . 3 (A V → (𝐹A) ⊆ ran 𝐹)
31, 2syl 14 . 2 (A 𝑊 → (𝐹A) ⊆ ran 𝐹)
4 rnexg 4540 . . 3 (𝐹 𝑉 → ran 𝐹 V)
5 uniexg 4141 . . 3 (ran 𝐹 V → ran 𝐹 V)
64, 5syl 14 . 2 (𝐹 𝑉 ran 𝐹 V)
7 ssexg 3887 . 2 (((𝐹A) ⊆ ran 𝐹 ran 𝐹 V) → (𝐹A) V)
83, 6, 7syl2anr 274 1 ((𝐹 𝑉 A 𝑊) → (𝐹A) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  wss 2911   cuni 3571  ran crn 4289  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-13 1401  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  ax-un 4136
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-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-cnv 4296  df-dm 4298  df-rn 4299  df-iota 4810  df-fv 4853
This theorem is referenced by:  fvex  5138  rdgivallem  5908  frecabex  5923  frecuzrdgrrn  8835  frec2uzrdg  8836  frecuzrdgrom  8837  frecuzrdgsuc  8842
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