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Theorem fvelrnb 5146
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 2290 . . . 4 (x A (𝐹x) = Bx(x A (𝐹x) = B))
2 19.41v 1764 . . . . 5 (x((x A (𝐹x) = B) 𝐹 Fn A) ↔ (x(x A (𝐹x) = B) 𝐹 Fn A))
3 ax-ia1 99 . . . . . . . . . 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 5117 . . . . . . . . 9 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
76funfni 4925 . . . . . . . 8 ((𝐹 Fn A x A) → (𝐹x) V)
85, 7syl 14 . . . . . . 7 (((x A (𝐹x) = B) 𝐹 Fn A) → (𝐹x) V)
9 ax-ia2 100 . . . . . . . . 9 ((x A (𝐹x) = B) → (𝐹x) = B)
109eleq1d 2088 . . . . . . . 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 1471 . . . . 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 5145 . . . 4 (𝐹 Fn A → ran 𝐹 = {yx A y = (𝐹x)})
1817eleq2d 2089 . . 3 (𝐹 Fn A → (B ran 𝐹B {yx A y = (𝐹x)}))
19 eqeq1 2028 . . . . . 6 (y = B → (y = (𝐹x) ↔ B = (𝐹x)))
20 eqcom 2024 . . . . . 6 (B = (𝐹x) ↔ (𝐹x) = B)
2119, 20syl6bb 185 . . . . 5 (y = B → (y = (𝐹x) ↔ (𝐹x) = B))
2221rexbidv 2305 . . . 4 (y = B → (x A y = (𝐹x) ↔ x A (𝐹x) = B))
2322elab3g 2670 . . 3 ((x A (𝐹x) = BB V) → (B {yx A y = (𝐹x)} ↔ x A (𝐹x) = B))
2418, 23sylan9bbr 439 . 2 (((x A (𝐹x) = BB V) 𝐹 Fn A) → (B ran 𝐹x A (𝐹x) = B))
2516, 24mpancom 401 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 1228  wex 1362   wcel 1374  {cab 2008  wrex 2285  Vcvv 2535  ran crn 4273   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  chfnrn  5203  rexrn  5229  ralrn  5230  elrnrexdmb  5232  ffnfv  5248  fconstfvm  5304  elunirn  5330  isoini  5382  reldm  5735
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