Theorem fvbr0 6921

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 2733	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
2		tz6.12i 6920	. . . 4 ⊢ ((𝐹‘𝑋) ≠ ∅ → ((𝐹‘𝑋) = (𝐹‘𝑋) → 𝑋𝐹(𝐹‘𝑋)))
3	1, 2	mpi 20	. . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋𝐹(𝐹‘𝑋))
4	3	necon1bi 2970	. 2 ⊢ (¬ 𝑋𝐹(𝐹‘𝑋) → (𝐹‘𝑋) = ∅)
5	4	orri 861	1 ⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)

Syntax hints:   ∨ wo 846   = wceq 1542   ≠ wne 2941  ∅c0 4323   class class class wbr 5149  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552

This theorem is referenced by:  fvrn0  6922  eliman0  6932