Theorem afvfv0bi 47872

Description: The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvfv0bi	⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Proof of Theorem afvfv0bi

Step	Hyp	Ref	Expression
1		ioran 999	. . . 4 ⊢ (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2		df-ne 2959	. . . . . . 7 ⊢ ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3		afvnufveq 47867	. . . . . . 7 ⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴))
4	2, 3	sylbir 238	. . . . . 6 ⊢ (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹‘𝐴))
5		eqeq1 2767	. . . . . . . 8 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹‘𝐴) = ∅))
6	5	notbid 321	. . . . . . 7 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → (¬ (𝐹'''𝐴) = ∅ ↔ ¬ (𝐹‘𝐴) = ∅))
7	6	biimpd 232	. . . . . 6 ⊢ ((𝐹'''𝐴) = (𝐹‘𝐴) → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹‘𝐴) = ∅))
8	4, 7	syl 18	. . . . 5 ⊢ (¬ (𝐹'''𝐴) = V → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹‘𝐴) = ∅))
9	8	impcom 412	. . . 4 ⊢ ((¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V) → ¬ (𝐹‘𝐴) = ∅)
10	1, 9	sylbi 220	. . 3 ⊢ (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → ¬ (𝐹‘𝐴) = ∅)
11	10	con4i 115	. 2 ⊢ ((𝐹‘𝐴) = ∅ → ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
12		afv0fv0 47869	. . 3 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)
13		afvpcfv0 47866	. . 3 ⊢ ((𝐹'''𝐴) = V → (𝐹‘𝐴) = ∅)
14	12, 13	jaoi 870	. 2 ⊢ (((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → (𝐹‘𝐴) = ∅)
15	11, 14	impbii 212	1 ⊢ ((𝐹‘𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ≠ wne 2958  Vcvv 3455  ∅c0 4287  ‘cfv 6538  '''cafv 47837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6494  df-fun 6540  df-fv 6546  df-aiota 47805  df-dfat 47839  df-afv 47840

This theorem is referenced by:  aovov0bi  47916