Theorem nelrnmpt 5948

Description: Non-membership in the range of a function in maps-to notation. (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 2960	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵)
4	3	ex 418	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵))
5	1, 4	ralrimi 3260	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵)
6		ralnex 3088	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
7	5, 6	sylib 221	. 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
8		nelrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
9		nelrnmpt.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt 5939	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
11	8, 10	syl 18	. 2 ⊢ (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
12	7, 11	mtbird 328	1 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ↦ cmpt 5185  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by:  deg1prod  34115