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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 2798 | . . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋) | |
2 | tz6.12i 6671 | . . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋))) | |
3 | 1, 2 | mpi 20 | . . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋𝐹(𝐹‘𝑋)) |
4 | 3 | necon1bi 3015 | . 2 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅) |
5 | 4 | orri 859 | 1 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1538 ≠ wne 2987 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 |
This theorem is referenced by: fvrn0 6673 eliman0 6680 |
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