Theorem afv2orxorb 44720

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

Step	Hyp	Ref	Expression
1		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝐵 = (𝐹''''𝐴) → (𝐵 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
2	1	eqcoms 2746	. . . . . . . . . 10 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
3	2	biimpa 477	. . . . . . . . 9 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
4		nnel 3058	. . . . . . . . 9 ⊢ (¬ (𝐹''''𝐴) ∉ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
5	3, 4	sylibr 233	. . . . . . . 8 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹)
6	5	a1d 25	. . . . . . 7 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
7		simpl 483	. . . . . . . 8 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → (𝐹''''𝐴) = 𝐵)
8	7	a1d 25	. . . . . . 7 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))
9	6, 8	jca 512	. . . . . 6 ⊢ (((𝐹''''𝐴) = 𝐵 ∧ 𝐵 ∈ ran 𝐹) → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
10	9	ex 413	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
11		eleq1 2826	. . . . . . . . . 10 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹))
12	11	anbi2d 629	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝐵 → (((𝐹''''𝐴) ∉ ran 𝐹 ∧ (𝐹''''𝐴) ∈ ran 𝐹) ↔ ((𝐹''''𝐴) ∉ ran 𝐹 ∧ 𝐵 ∈ ran 𝐹)))
13		elnelall 3062	. . . . . . . . . 10 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
14	13	impcom 408	. . . . . . . . 9 ⊢ (((𝐹''''𝐴) ∉ ran 𝐹 ∧ (𝐹''''𝐴) ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹)
15	12, 14	syl6bir 253	. . . . . . . 8 ⊢ ((𝐹''''𝐴) = 𝐵 → (((𝐹''''𝐴) ∉ ran 𝐹 ∧ 𝐵 ∈ ran 𝐹) → ¬ (𝐹''''𝐴) ∉ ran 𝐹))
16	15	com12 32	. . . . . . 7 ⊢ (((𝐹''''𝐴) ∉ ran 𝐹 ∧ 𝐵 ∈ 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))
18	17	adantr 481	. . . . . . 7 ⊢ (((𝐹''''𝐴) ∉ ran 𝐹 ∧ 𝐵 ∈ ran 𝐹) → (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))
19	16, 18	jca 512	. . . . . 6 ⊢ (((𝐹''''𝐴) ∉ ran 𝐹 ∧ 𝐵 ∈ ran 𝐹) → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
20	19	ex 413	. . . . 5 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵))))
21	10, 20	jaoi 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))))
22	21	com12 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 1507	. . . 4 ⊢ (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ¬ ((𝐹''''𝐴) = 𝐵 ↔ (𝐹''''𝐴) ∉ ran 𝐹))
24		xor3 384	. . . 4 ⊢ (¬ ((𝐹''''𝐴) = 𝐵 ↔ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ↔ ¬ (𝐹''''𝐴) ∉ ran 𝐹))
25		dfbi2 475	. . . 4 ⊢ (((𝐹''''𝐴) = 𝐵 ↔ ¬ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
26	23, 24, 25	3bitri 297	. . 3 ⊢ (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) ↔ (((𝐹''''𝐴) = 𝐵 → ¬ (𝐹''''𝐴) ∉ ran 𝐹) ∧ (¬ (𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝐵)))
27	22, 26	syl6ibr 251	. 2 ⊢ (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))
28		xoror 1514	. 2 ⊢ (((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹) → ((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹))
29	27, 28	impbid1 224	1 ⊢ (𝐵 ∈ ran 𝐹 → (((𝐹''''𝐴) = 𝐵 ∨ (𝐹''''𝐴) ∉ ran 𝐹) ↔ ((𝐹''''𝐴) = 𝐵 ⊻ (𝐹''''𝐴) ∉ ran 𝐹)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ⊻ wxo 1506   = wceq 1539   ∈ wcel 2106   ∉ wnel 3049  ran crn 5590  ''''cafv2 44700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-xor 1507  df-ex 1783  df-cleq 2730  df-clel 2816  df-nel 3050

This theorem is referenced by:  afv2fv0xorb  44759