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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 2787 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵) |
4 | 3 | ex 449 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵)) |
5 | 1, 4 | ralrimi 2940 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵) |
6 | ralnex 2975 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) | |
7 | 5, 6 | sylib 207 | . 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
8 | nelrnmpt.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | nelrnmpt.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | elrnmpt 5293 | . . 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 314 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: (None) |
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