Theorem 0fv 5664

Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Assertion

Ref	Expression
0fv	⊢ (∅‘𝐴) = ∅

Proof of Theorem 0fv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5325	. 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥)
2		noel 3495	. . . . . 6 ⊢ ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3		df-br 4083	. . . . . 6 ⊢ (𝐴∅𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
4	2, 3	mtbir 675	. . . . 5 ⊢ ¬ 𝐴∅𝑥
5	4	nex 1546	. . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥
6		euex 2107	. . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥)
7	5, 6	mto 666	. . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥
8		iotanul 5293	. . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅)
9	7, 8	ax-mp 5	. 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅
10	1, 9	eqtri 2250	1 ⊢ (∅‘𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1395  ∃wex 1538  ∃!weu 2077   ∈ wcel 2200  ∅c0 3491  ⟨cop 3669   class class class wbr 4082  ℩cio 5275  ‘cfv 5317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325

This theorem is referenced by:  fv2prc  5665  ccat1st1st  11167  strsl0  13076