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Theorem fvelrnb 5164
 Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
Assertion
Ref Expression
fvelrnb (𝐹 Fn A → (B ran 𝐹x A (𝐹x) = B))
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem fvelrnb
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . . . 4 (x A (𝐹x) = Bx(x A (𝐹x) = B))
2 19.41v 1779 . . . . 5 (x((x A (𝐹x) = B) 𝐹 Fn A) ↔ (x(x A (𝐹x) = B) 𝐹 Fn A))
3 simpl 102 . . . . . . . . . 10 ((x A (𝐹x) = B) → x A)
43anim1i 323 . . . . . . . . 9 (((x A (𝐹x) = B) 𝐹 Fn A) → (x A 𝐹 Fn A))
54ancomd 254 . . . . . . . 8 (((x A (𝐹x) = B) 𝐹 Fn A) → (𝐹 Fn A x A))
6 funfvex 5135 . . . . . . . . 9 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
76funfni 4942 . . . . . . . 8 ((𝐹 Fn A x A) → (𝐹x) V)
85, 7syl 14 . . . . . . 7 (((x A (𝐹x) = B) 𝐹 Fn A) → (𝐹x) V)
9 simpr 103 . . . . . . . . 9 ((x A (𝐹x) = B) → (𝐹x) = B)
109eleq1d 2103 . . . . . . . 8 ((x A (𝐹x) = B) → ((𝐹x) V ↔ B V))
1110adantr 261 . . . . . . 7 (((x A (𝐹x) = B) 𝐹 Fn A) → ((𝐹x) V ↔ B V))
128, 11mpbid 135 . . . . . 6 (((x A (𝐹x) = B) 𝐹 Fn A) → B V)
1312exlimiv 1486 . . . . 5 (x((x A (𝐹x) = B) 𝐹 Fn A) → B V)
142, 13sylbir 125 . . . 4 ((x(x A (𝐹x) = B) 𝐹 Fn A) → B V)
151, 14sylanb 268 . . 3 ((x A (𝐹x) = B 𝐹 Fn A) → B V)
1615expcom 109 . 2 (𝐹 Fn A → (x A (𝐹x) = BB V))
17 fnrnfv 5163 . . . 4 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
1817eleq2d 2104 . . 3 (𝐹 Fn A → (B ran 𝐹B {yx A y = (𝐹x)}))
19 eqeq1 2043 . . . . . 6 (y = B → (y = (𝐹x) ↔ B = (𝐹x)))
20 eqcom 2039 . . . . . 6 (B = (𝐹x) ↔ (𝐹x) = B)
2119, 20syl6bb 185 . . . . 5 (y = B → (y = (𝐹x) ↔ (𝐹x) = B))
2221rexbidv 2321 . . . 4 (y = B → (x A y = (𝐹x) ↔ x A (𝐹x) = B))
2322elab3g 2687 . . 3 ((x A (𝐹x) = BB V) → (B {yx A y = (𝐹x)} ↔ x A (𝐹x) = B))
2418, 23sylan9bbr 436 . 2 (((x A (𝐹x) = BB V) 𝐹 Fn A) → (B ran 𝐹x A (𝐹x) = B))
2516, 24mpancom 399 1 (𝐹 Fn A → (B ran 𝐹x A (𝐹x) = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551  ran crn 4289   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-bndl 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:  chfnrn  5221  rexrn  5247  ralrn  5248  elrnrexdmb  5250  ffnfv  5266  fconstfvm  5322  elunirn  5348  isoini  5400  reldm  5754  uzn0  8264  frec2uzrand  8872  frecuzrdgfn  8879
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