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Theorem nelrnmpt 38761
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 2795 . . . . 5 ((𝜑𝑥𝐴) → ¬ 𝐶 = 𝐵)
43ex 450 . . . 4 (𝜑 → (𝑥𝐴 → ¬ 𝐶 = 𝐵))
51, 4ralrimi 2951 . . 3 (𝜑 → ∀𝑥𝐴 ¬ 𝐶 = 𝐵)
6 ralnex 2986 . . 3 (∀𝑥𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥𝐴 𝐶 = 𝐵)
75, 6sylib 208 . 2 (𝜑 → ¬ ∃𝑥𝐴 𝐶 = 𝐵)
8 nelrnmpt.c . . 3 (𝜑𝐶𝑉)
9 nelrnmpt.f . . . 4 𝐹 = (𝑥𝐴𝐵)
109elrnmpt 5334 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
118, 10syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
127, 11mtbird 315 1 (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  wcel 1987  wne 2790  wral 2907  wrex 2908  cmpt 4675  ran crn 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-mpt 4677  df-cnv 5084  df-dm 5086  df-rn 5087
This theorem is referenced by: (None)
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