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| Mirrors > Home > MPE Home > Th. List > fvbr0 | Structured version Visualization version GIF version | ||
| Description: Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| fvbr0 | ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋) | |
| 2 | tz6.12i 6934 | . . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))) | |
| 3 | 1, 2 | mpi 20 | . . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋𝐹(𝐹‘𝑋)) | 
| 4 | 3 | necon1bi 2969 | . 2 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) | 
| 5 | 4 | orri 863 | 1 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∨ wo 848 = wceq 1540 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: fvrn0 6936 eliman0 6946 | 
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