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Theorem elrnmptg 4761
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 4757 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
32eleq2i 2184 . 2 (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
4 r19.29 2546 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → ∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵))
5 eleq1 2180 . . . . . . . 8 (𝐶 = 𝐵 → (𝐶𝑉𝐵𝑉))
65biimparc 297 . . . . . . 7 ((𝐵𝑉𝐶 = 𝐵) → 𝐶𝑉)
7 elex 2671 . . . . . . 7 (𝐶𝑉𝐶 ∈ V)
86, 7syl 14 . . . . . 6 ((𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
98rexlimivw 2522 . . . . 5 (∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
104, 9syl 14 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
1110ex 114 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V))
12 eqeq1 2124 . . . . 5 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
1312rexbidv 2415 . . . 4 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1413elab3g 2808 . . 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 191 1 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  cmpt 3959  ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  elrnmpti  4762  fliftel  5662
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