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Theorem fvelrnb 5273
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 𝐹 ↔ ∃𝑥𝐴 (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 2359 . . . 4 (∃𝑥𝐴 (𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵))
2 19.41v 1825 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3 simpl 107 . . . . . . . . . 10 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → 𝑥𝐴)
43anim1i 333 . . . . . . . . 9 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥𝐴𝐹 Fn 𝐴))
54ancomd 263 . . . . . . . 8 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
6 funfvex 5243 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
76funfni 5050 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
85, 7syl 14 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹𝑥) ∈ V)
9 simpr 108 . . . . . . . . 9 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → (𝐹𝑥) = 𝐵)
109eleq1d 2151 . . . . . . . 8 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
1110adantr 270 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
128, 11mpbid 145 . . . . . 6 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
1312exlimiv 1530 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
142, 13sylbir 133 . . . 4 ((∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
151, 14sylanb 278 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐹 Fn 𝐴) → 𝐵 ∈ V)
1615expcom 114 . 2 (𝐹 Fn 𝐴 → (∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V))
17 fnrnfv 5272 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1817eleq2d 2152 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}))
19 eqeq1 2089 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ 𝐵 = (𝐹𝑥)))
20 eqcom 2085 . . . . . 6 (𝐵 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵)
2119, 20syl6bb 194 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵))
2221rexbidv 2374 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2322elab3g 2752 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2418, 23sylan9bbr 451 . 2 (((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2516, 24mpancom 413 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354  Vcvv 2610  ran crn 4392   Fn wfn 4947  cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  chfnrn  5330  rexrn  5356  ralrn  5357  elrnrexdmb  5359  ffnfv  5375  fconstfvm  5431  elunirn  5457  isoini  5508  reldm  5863  ordiso2  6540  uzn0  8767  frec2uzrand  9539  frecuzrdgtcl  9546  frecuzrdgfunlem  9553  uzin2  10074
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