Theorem 0fv 5728

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 5380	. 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥)
2		noel 3525	. . . . . 6 ⊢ ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3		df-br 4126	. . . . . 6 ⊢ (𝐴∅𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
4	2, 3	mtbir 682	. . . . 5 ⊢ ¬ 𝐴∅𝑥
5	4	nex 1553	. . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥
6		euex 2116	. . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥)
7	5, 6	mto 672	. . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥
8		iotanul 5348	. . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅)
9	7, 8	ax-mp 5	. 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅
10	1, 9	eqtri 2259	1 ⊢ (∅‘𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1402  ∃wex 1545  ∃!weu 2086   ∈ wcel 2209  ∅c0 3520  ⟨cop 3708   class class class wbr 4125  ℩cio 5330  ‘cfv 5372

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  fv2prc  5729  ccat1st1st  11387  strsl0  13379