Theorem afvfv0bi 47745

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 2961	. . . . . . 7 ⊢ ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3		afvnufveq 47740	. . . . . . 7 ⊢ ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹‘𝐴))
4	2, 3	sylbir 238	. . . . . 6 ⊢ (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹‘𝐴))
5		eqeq1 2769	. . . . . . . 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 47742	. . 3 ⊢ ((𝐹'''𝐴) = ∅ → (𝐹‘𝐴) = ∅)
13		afvpcfv0 47739	. . 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 1563   ≠ wne 2960  Vcvv 3457  ∅c0 4288  ‘cfv 6525  '''cafv 47710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-aiota 47678  df-dfat 47712  df-afv 47713

This theorem is referenced by:  aovov0bi  47789