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Theorem elrnmptg 5283
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 5279 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
32eleq2i 2679 . 2 (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
4 r19.29 3053 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → ∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵))
5 eleq1 2675 . . . . . . . 8 (𝐶 = 𝐵 → (𝐶𝑉𝐵𝑉))
65biimparc 502 . . . . . . 7 ((𝐵𝑉𝐶 = 𝐵) → 𝐶𝑉)
76elexd 3186 . . . . . 6 ((𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
87rexlimivw 3010 . . . . 5 (∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
94, 8syl 17 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
109ex 448 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V))
11 eqeq1 2613 . . . . 5 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
1211rexbidv 3033 . . . 4 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1312elab3g 3325 . . 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 270 1 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wrex 2896  Vcvv 3172  cmpt 4637  ran crn 5029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-cnv 5036  df-dm 5038  df-rn 5039
This theorem is referenced by:  elrnmpti  5284
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