Theorem 0fv 5713

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 5365	. 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥)
2		noel 3516	. . . . . 6 ⊢ ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3		df-br 4115	. . . . . 6 ⊢ (𝐴∅𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
4	2, 3	mtbir 678	. . . . 5 ⊢ ¬ 𝐴∅𝑥
5	4	nex 1549	. . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥
6		euex 2112	. . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥)
7	5, 6	mto 668	. . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥
8		iotanul 5333	. . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅)
9	7, 8	ax-mp 5	. 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅
10	1, 9	eqtri 2255	1 ⊢ (∅‘𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1398  ∃wex 1541  ∃!weu 2082   ∈ wcel 2205  ∅c0 3512  ⟨cop 3697   class class class wbr 4114  ℩cio 5315  ‘cfv 5357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  fv2prc  5714  ccat1st1st  11354  strsl0  13345