Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afv2orxorb Structured version   Visualization version   GIF version

Theorem afv2orxorb 44180
 Description: If a set is in the range of a function, the alternate function value at a class 𝐴 equals this set or is not in the range of the function iff the alternate function value at the class 𝐴 either equals this set or is not in the range of the function. If 𝐵 ∉ ran 𝐹, both disjuncts of the exclusive or can be true: (𝐹''''𝐴) = 𝐵 → (𝐹''''𝐴) ∉ ran 𝐹. (Contributed by AV, 11-Sep-2022.)
Assertion
Ref Expression
afv2orxorb (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

Proof of Theorem afv2orxorb
StepHypRef Expression
1 eleq1 2839 . . . . . . . . . . 11 (𝐵 = (𝐹''''𝐴) → (𝐵 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
21eqcoms 2766 . . . . . . . . . 10 ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
32biimpa 480 . . . . . . . . 9 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
4 nnel 3064 . . . . . . . . 9 (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
53, 4sylibr 237 . . . . . . . 8 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹)
65a1d 25 . . . . . . 7 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
7 simpl 486 . . . . . . . 8 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → (𝐹''''𝐴) = 𝐵)
87a1d 25 . . . . . . 7 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))
96, 8jca 515 . . . . . 6 (((𝐹''''𝐴) = 𝐵𝐵 ∈ ran 𝐹) → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
109ex 416 . . . . 5 ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
11 eleq1 2839 . . . . . . . . . 10 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
1211anbi2d 631 . . . . . . . . 9 ((𝐹''''𝐴) = 𝐵 → (((𝐹''''𝐴) ∉ ran 𝐹 ∧ (𝐹''''𝐴) ∈ ran 𝐹) ↔ ((𝐹''''𝐴) ∉ ran 𝐹𝐵 ∈ ran 𝐹)))
13 elnelall 3068 . . . . . . . . . 10 ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
1413impcom 411 . . . . . . . . 9 (((𝐹''''𝐴) ∉ ran 𝐹 ∧ (𝐹''''𝐴) ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹)
1512, 14syl6bir 257 . . . . . . . 8 ((𝐹''''𝐴) = 𝐵 → (((𝐹''''𝐴) ∉ ran 𝐹𝐵 ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
1615com12 32 . . . . . . 7 (((𝐹''''𝐴) ∉ ran 𝐹𝐵 ∈ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
17 pm2.24 124 . . . . . . . 8 ((𝐹''''𝐴) ∉ ran 𝐹 → (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))
1817adantr 484 . . . . . . 7 (((𝐹''''𝐴) ∉ ran 𝐹𝐵 ∈ ran 𝐹) → (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))
1916, 18jca 515 . . . . . 6 (((𝐹''''𝐴) ∉ ran 𝐹𝐵 ∈ ran 𝐹) → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
2019ex 416 . . . . 5 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
2110, 20jaoi 854 . . . 4 (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
2221com12 32 . . 3 (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
23 df-xor 1503 . . . 4 (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ¬ ((𝐹''''𝐴) = 𝐵 ↔ (𝐹''''𝐴) ∉ ran 𝐹))
24 xor3 387 . . . 4 (¬ ((𝐹''''𝐴) = 𝐵 ↔ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ↔ ¬ (𝐹''''𝐴) ∉ ran 𝐹))
25 dfbi2 478 . . . 4 (((𝐹''''𝐴) = 𝐵 ↔ ¬ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
2623, 24, 253bitri 300 . . 3 (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
2722, 26syl6ibr 255 . 2 (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
28 xoror 1510 . 2 (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹))
2927, 28impbid1 228 1 (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ⊻ wxo 1502   = wceq 1538   ∈ wcel 2111   ∉ wnel 3055  ran crn 5528  ''''cafv2 44160 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-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-xor 1503  df-ex 1782  df-cleq 2750  df-clel 2830  df-nel 3056 This theorem is referenced by:  afv2fv0xorb  44219
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