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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 3023 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵) |
4 | 3 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵)) |
5 | 1, 4 | ralrimi 3218 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵) |
6 | ralnex 3238 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) | |
7 | 5, 6 | sylib 220 | . 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
8 | nelrnmpt.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
9 | nelrnmpt.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | elrnmpt 5830 | . . 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 327 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ↦ cmpt 5148 ran crn 5558 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: (None) |
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