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Mirrors > Home > MPE Home > Th. List > Mathboxes > nelrnmpt | Structured version Visualization version GIF version |
Description: Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
nelrnmpt.x | ⊢ Ⅎ𝑥𝜑 |
nelrnmpt.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
nelrnmpt.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
nelrnmpt.n | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 𝐵) |
Ref | Expression |
---|---|
nelrnmpt | ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹) |
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) |
Colors of variables: wff setvar class |
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) |
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