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	⊢ Ⅎ𝑥𝜑
nelrnmpt.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
nelrnmpt.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
nelrnmpt.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 𝐵)

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

Step	Hyp	Ref	Expression
1		nelrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nelrnmpt.n	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 𝐵)
3	2	neneqd 2973	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵)
4	3	ex 405	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵))
5	1, 4	ralrimi 3167	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵)
6		ralnex 3184	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	5, 6	sylib 210	. 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		nelrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
9		nelrnmpt.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt 5671	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
11	8, 10	syl 17	. 2 ⊢ (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
12	7, 11	mtbird 317	1 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  Ⅎwnf 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)