Theorem 0fv 5410

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 5089	. 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥)
2		noel 3333	. . . . . 6 ⊢ ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3		df-br 3896	. . . . . 6 ⊢ (𝐴∅𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
4	2, 3	mtbir 643	. . . . 5 ⊢ ¬ 𝐴∅𝑥
5	4	nex 1459	. . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥
6		euex 2005	. . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥)
7	5, 6	mto 634	. . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥
8		iotanul 5061	. . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅)
9	7, 8	ax-mp 7	. 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅
10	1, 9	eqtri 2135	1 ⊢ (∅‘𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1314  ∃wex 1451   ∈ wcel 1463  ∃!weu 1975  ∅c0 3329  ⟨cop 3496   class class class wbr 3895  ℩cio 5044  ‘cfv 5081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-nul 3330  df-sn 3499  df-uni 3703  df-br 3896  df-iota 5046  df-fv 5089

This theorem is referenced by:  strsl0  11850