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Mathbox for Glauco Siliprandi |
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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 3004 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝐶 = 𝐵) |
4 | 3 | ex 403 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ 𝐶 = 𝐵)) |
5 | 1, 4 | ralrimi 3166 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝐶 = 𝐵) |
6 | ralnex 3201 | . . 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 5605 | . . 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 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 ↦ cmpt 4952 ran crn 5343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-mpt 4953 df-cnv 5350 df-dm 5352 df-rn 5353 |
This theorem is referenced by: (None) |
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