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Theorem elrnmptg 5530
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 5526 . . 3 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
32eleq2i 2831 . 2 (𝐶 ∈ ran 𝐹𝐶 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
4 r19.29 3210 . . . . 5 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → ∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵))
5 eleq1 2827 . . . . . . . 8 (𝐶 = 𝐵 → (𝐶𝑉𝐵𝑉))
65biimparc 505 . . . . . . 7 ((𝐵𝑉𝐶 = 𝐵) → 𝐶𝑉)
76elexd 3354 . . . . . 6 ((𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
87rexlimivw 3167 . . . . 5 (∃𝑥𝐴 (𝐵𝑉𝐶 = 𝐵) → 𝐶 ∈ V)
94, 8syl 17 . . . 4 ((∀𝑥𝐴 𝐵𝑉 ∧ ∃𝑥𝐴 𝐶 = 𝐵) → 𝐶 ∈ V)
109ex 449 . . 3 (∀𝑥𝐴 𝐵𝑉 → (∃𝑥𝐴 𝐶 = 𝐵𝐶 ∈ V))
11 eqeq1 2764 . . . . 5 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
1211rexbidv 3190 . . . 4 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
1312elab3g 3497 . . 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 272 1 (∀𝑥𝐴 𝐵𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  elrnmpti  5531
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