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Theorem fnfvima 5420
Description: The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
Assertion
Ref Expression
fnfvima ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))

Proof of Theorem fnfvima
StepHypRef Expression
1 fnfun 5023 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 936 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → Fun 𝐹)
3 simp2 916 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆𝐴)
4 fndm 5025 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
543ad2ant1 936 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → dom 𝐹 = 𝐴)
63, 5sseqtr4d 3009 . . 3 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑆 ⊆ dom 𝐹)
72, 6jca 294 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (Fun 𝐹𝑆 ⊆ dom 𝐹))
8 simp3 917 . 2 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → 𝑋𝑆)
9 funfvima2 5418 . 2 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → (𝑋𝑆 → (𝐹𝑋) ∈ (𝐹𝑆)))
107, 8, 9sylc 60 1 ((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  wss 2944  dom cdm 4372  cima 4375  Fun wfun 4923   Fn wfn 4924  cfv 4929
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by: (None)
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