Theorem nelrnmpt 45180

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 2933	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵)
4	3	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵))
5	1, 4	ralrimi 3230	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵)
6		ralnex 3058	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	5, 6	sylib 218	. 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		nelrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
9		nelrnmpt.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt 5897	. . 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 325	1 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ↦ cmpt 5170  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by: (None)