Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋) | |
2 | tz6.12i 6689 | . . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))) | |
3 | 1, 2 | mpi 20 | . . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋𝐹(𝐹‘𝑋)) |
4 | 3 | necon1bi 3041 | . 2 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
5 | 4 | orri 856 | 1 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 841 = wceq 1528 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 |
This theorem is referenced by: fvrn0 6691 eliman0 6698 |
Copyright terms: Public domain | W3C validator |