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Theorem fvelrnb 5584
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 2474 . . . 4 (∃𝑥𝐴 (𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵))
2 19.41v 1914 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) ↔ (∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴))
3 simpl 109 . . . . . . . . . 10 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → 𝑥𝐴)
43anim1i 340 . . . . . . . . 9 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝑥𝐴𝐹 Fn 𝐴))
54ancomd 267 . . . . . . . 8 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹 Fn 𝐴𝑥𝐴))
6 funfvex 5551 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
76funfni 5335 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
85, 7syl 14 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → (𝐹𝑥) ∈ V)
9 simpr 110 . . . . . . . . 9 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → (𝐹𝑥) = 𝐵)
109eleq1d 2258 . . . . . . . 8 ((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
1110adantr 276 . . . . . . 7 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → ((𝐹𝑥) ∈ V ↔ 𝐵 ∈ V))
128, 11mpbid 147 . . . . . 6 (((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
1312exlimiv 1609 . . . . 5 (∃𝑥((𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
142, 13sylbir 135 . . . 4 ((∃𝑥(𝑥𝐴 ∧ (𝐹𝑥) = 𝐵) ∧ 𝐹 Fn 𝐴) → 𝐵 ∈ V)
151, 14sylanb 284 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐹 Fn 𝐴) → 𝐵 ∈ V)
1615expcom 116 . 2 (𝐹 Fn 𝐴 → (∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V))
17 fnrnfv 5583 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1817eleq2d 2259 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}))
19 eqeq1 2196 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ 𝐵 = (𝐹𝑥)))
20 eqcom 2191 . . . . . 6 (𝐵 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵)
2119, 20bitrdi 196 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝐵))
2221rexbidv 2491 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2322elab3g 2903 . . 3 ((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2418, 23sylan9bbr 463 . 2 (((∃𝑥𝐴 (𝐹𝑥) = 𝐵𝐵 ∈ V) ∧ 𝐹 Fn 𝐴) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
2516, 24mpancom 422 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wrex 2469  Vcvv 2752  ran crn 4645   Fn wfn 5230  cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  foelcdmi  5589  chfnrn  5648  rexrn  5674  ralrn  5675  elrnrexdmb  5677  ffnfv  5695  fconstfvm  5755  elunirn  5788  isoini  5840  canth  5850  reldm  6212  ordiso2  7065  eldju  7098  ctssdc  7143  uzn0  9575  frec2uzrand  10438  frecuzrdgtcl  10445  frecuzrdgfunlem  10452  uzin2  11031  imasgrp2  13067  imasrng  13327  imasring  13431  reeff1o  14671
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