Theorem afv2fv0xorb 48552

Description: If a set is in the range of a function, the function's value at an argument is the empty set if and only if the alternate function value at this argument is either the empty set or undefined. (Contributed by AV, 11-Sep-2022.)

Assertion

Ref	Expression
afv2fv0xorb	⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

Proof of Theorem afv2fv0xorb

Step	Hyp	Ref	Expression
1		afv2fv0b 48551	. 2 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
2		afv2orxorb 48513	. 2 ⊢ (∅ ∈ ran 𝐹 → (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
3	1, 2	bitrid 286	1 ⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   ⊻ wxo 1541   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  ∅c0 4281  ran crn 5657  ‘cfv 6561  ''''cafv2 48493

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fv 6569  df-dfat 48405  df-afv2 48494

This theorem is used by: (None)