Theorem afv2fv0xorb 47825

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 47824	. 2 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹))
2		afv2orxorb 47786	. 2 ⊢ (∅ ∈ ran 𝐹 → (((𝐹''''𝐴) = ∅ ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
3	1, 2	bitrid 285	1 ⊢ (∅ ∈ ran 𝐹 → ((𝐹‘𝐴) = ∅ ↔ ((𝐹''''𝐴) = ∅ ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 858   ⊻ wxo 1530   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  ∅c0 4285  ran crn 5646  ‘cfv 6517  ''''cafv2 47766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-xor 1531  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-iota 6473  df-fun 6519  df-fv 6525  df-dfat 47677  df-afv2 47767

This theorem is referenced by: (None)