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Mirrors > Home > MPE Home > Th. List > nf1oconst | Structured version Visualization version GIF version |
Description: A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.) |
Ref | Expression |
---|---|
nf1oconst | ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1-onto→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf1const 7319 | . . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶) | |
2 | 1 | orcd 871 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (¬ 𝐹:𝐴–1-1→𝐶 ∨ ¬ 𝐹:𝐴–onto→𝐶)) |
3 | ianor 979 | . . 3 ⊢ (¬ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (¬ 𝐹:𝐴–1-1→𝐶 ∨ ¬ 𝐹:𝐴–onto→𝐶)) | |
4 | df-f1o 6560 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶)) | |
5 | 3, 4 | xchnxbir 332 | . 2 ⊢ (¬ 𝐹:𝐴–1-1-onto→𝐶 ↔ (¬ 𝐹:𝐴–1-1→𝐶 ∨ ¬ 𝐹:𝐴–onto→𝐶)) |
6 | 2, 5 | sylibr 233 | 1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1-onto→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2937 {csn 4632 ⟶wf 6549 –1-1→wf1 6550 –onto→wfo 6551 –1-1-onto→wf1o 6552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-f1o 6560 df-fv 6561 |
This theorem is referenced by: symgpssefmnd 19364 |
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