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Theorem nelrnmpt 38283
 Description: Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
nelrnmpt.x 𝑥𝜑
nelrnmpt.f 𝐹 = (𝑥𝐴𝐵)
nelrnmpt.c (𝜑𝐶𝑉)
nelrnmpt.n ((𝜑𝑥𝐴) → 𝐶𝐵)
Assertion
Ref Expression
nelrnmpt (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem nelrnmpt
StepHypRef Expression
1 nelrnmpt.x . . . 4 𝑥𝜑
2 nelrnmpt.n . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
32neneqd 2787 . . . . 5 ((𝜑𝑥𝐴) → ¬ 𝐶 = 𝐵)
43ex 449 . . . 4 (𝜑 → (𝑥𝐴 → ¬ 𝐶 = 𝐵))
51, 4ralrimi 2940 . . 3 (𝜑 → ∀𝑥𝐴 ¬ 𝐶 = 𝐵)
6 ralnex 2975 . . 3 (∀𝑥𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥𝐴 𝐶 = 𝐵)
75, 6sylib 207 . 2 (𝜑 → ¬ ∃𝑥𝐴 𝐶 = 𝐵)
8 nelrnmpt.c . . 3 (𝜑𝐶𝑉)
9 nelrnmpt.f . . . 4 𝐹 = (𝑥𝐴𝐵)
109elrnmpt 5293 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
118, 10syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
127, 11mtbird 314 1 (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ↦ cmpt 4643  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by: (None)
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