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Theorem elrnmptg 5828
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 5824 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
32eleq2i 2829 . 2 (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
4 r19.29 3176 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → ∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵))
5 eleq1 2825 . . . . . . . 8 (𝐶 = 𝐵 → (𝐶𝑉𝐵𝑉))
65biimparc 483 . . . . . . 7 ((𝐵𝑉𝐶 = 𝐵) → 𝐶𝑉)
76elexd 3428 . . . . . 6 ((𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
87rexlimivw 3201 . . . . 5 (∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
94, 8syl 17 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
109ex 416 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V))
11 eqeq1 2741 . . . . 5 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
1211rexbidv 3216 . . . 4 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1312elab3g 3594 . . 3 ((∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1410, 13syl 17 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ↔ ∃𝑥𝐴 𝐶 = 𝐵))
153, 14syl5bb 286 1 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062  Vcvv 3408  cmpt 5135  ran crn 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-mpt 5136  df-cnv 5559  df-dm 5561  df-rn 5562
This theorem is referenced by:  elrnmpti  5829  iunrnmptss  30624
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