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Theorem fvelrnb 5476
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 2423 . . . 4 (∃𝑥𝐴 (𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵))
2 19.41v 1875 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3 simpl 108 . . . . . . . . . 10 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → 𝑥𝐴)
43anim1i 338 . . . . . . . . 9 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥𝐴𝐹 Fn 𝐴))
54ancomd 265 . . . . . . . 8 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
6 funfvex 5445 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
76funfni 5230 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
85, 7syl 14 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹𝑥) ∈ V)
9 simpr 109 . . . . . . . . 9 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → (𝐹𝑥) = 𝐵)
109eleq1d 2209 . . . . . . . 8 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
1110adantr 274 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
128, 11mpbid 146 . . . . . 6 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
1312exlimiv 1578 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
142, 13sylbir 134 . . . 4 ((∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
151, 14sylanb 282 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐹 Fn 𝐴) → 𝐵 ∈ V)
1615expcom 115 . 2 (𝐹 Fn 𝐴 → (∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V))
17 fnrnfv 5475 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1817eleq2d 2210 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}))
19 eqeq1 2147 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ 𝐵 = (𝐹𝑥)))
20 eqcom 2142 . . . . . 6 (𝐵 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵)
2119, 20syl6bb 195 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵))
2221rexbidv 2439 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2322elab3g 2838 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2418, 23sylan9bbr 459 . 2 (((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2516, 24mpancom 419 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  {cab 2126  wrex 2418  Vcvv 2689  ran crn 4547   Fn wfn 5125  cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138
This theorem is referenced by:  chfnrn  5538  rexrn  5564  ralrn  5565  elrnrexdmb  5567  ffnfv  5585  fconstfvm  5645  elunirn  5674  isoini  5726  reldm  6091  ordiso2  6927  eldju  6960  ctssdc  7005  uzn0  9364  frec2uzrand  10208  frecuzrdgtcl  10215  frecuzrdgfunlem  10222  uzin2  10790  reeff1o  12900
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