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Theorem elrnmptg 4687
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
elrnmptg (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
21rnmpt 4683 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
32eleq2i 2154 . 2 (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
4 r19.29 2506 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → ∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵))
5 eleq1 2150 . . . . . . . 8 (𝐶 = 𝐵 → (𝐶𝑉𝐵𝑉))
65biimparc 293 . . . . . . 7 ((𝐵𝑉𝐶 = 𝐵) → 𝐶𝑉)
7 elex 2630 . . . . . . 7 (𝐶𝑉𝐶 ∈ V)
86, 7syl 14 . . . . . 6 ((𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
98rexlimivw 2485 . . . . 5 (∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
104, 9syl 14 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
1110ex 113 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V))
12 eqeq1 2094 . . . . 5 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
1312rexbidv 2381 . . . 4 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1413elab3g 2766 . . 3 ((∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1511, 14syl 14 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝐶 = 𝐵))
163, 15syl5bb 190 1 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  cmpt 3899  ran crn 4439
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  elrnmpti  4688  fliftel  5572
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