Theorem fvbr0 6936

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.)

Assertion

Ref	Expression
fvbr0	⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)

Proof of Theorem fvbr0

Step	Hyp	Ref	Expression
1		eqid 2735	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
2		tz6.12i 6935	. . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋)))
3	1, 2	mpi 20	. . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋𝐹(𝐹‘𝑋))
4	3	necon1bi 2967	. 2 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
5	4	orri 862	1 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)

Syntax hints:   ∨ wo 847   = wceq 1537   ≠ wne 2938  ∅c0 4339   class class class wbr 5148  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571

This theorem is referenced by:  fvrn0  6937  eliman0  6947