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Theorem nelrnmpt 40074
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 3004 . . . . 5 ((𝜑𝑥𝐴) → ¬ 𝐶 = 𝐵)
43ex 403 . . . 4 (𝜑 → (𝑥𝐴 → ¬ 𝐶 = 𝐵))
51, 4ralrimi 3166 . . 3 (𝜑 → ∀𝑥𝐴 ¬ 𝐶 = 𝐵)
6 ralnex 3201 . . 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 5605 . . 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 386   = wceq 1658  wnf 1884  wcel 2166  wne 2999  wral 3117  wrex 3118  cmpt 4952  ran crn 5343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-mpt 4953  df-cnv 5350  df-dm 5352  df-rn 5353
This theorem is referenced by: (None)
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