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Theorem nelrnmpt 40765
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 2973 . . . . 5 ((𝜑𝑥𝐴) → ¬ 𝐶 = 𝐵)
43ex 405 . . . 4 (𝜑 → (𝑥𝐴 → ¬ 𝐶 = 𝐵))
51, 4ralrimi 3167 . . 3 (𝜑 → ∀𝑥𝐴 ¬ 𝐶 = 𝐵)
6 ralnex 3184 . . 3 (∀𝑥𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥𝐴 𝐶 = 𝐵)
75, 6sylib 210 . 2 (𝜑 → ¬ ∃𝑥𝐴 𝐶 = 𝐵)
8 nelrnmpt.c . . 3 (𝜑𝐶𝑉)
9 nelrnmpt.f . . . 4 𝐹 = (𝑥𝐴𝐵)
109elrnmpt 5671 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
118, 10syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
127, 11mtbird 317 1 (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wnf 1746  wcel 2050  wne 2968  wral 3089  wrex 3090  cmpt 5008  ran crn 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-mpt 5009  df-cnv 5415  df-dm 5417  df-rn 5418
This theorem is referenced by: (None)
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