Theorem 0fv 5546

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 5220	. 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥)
2		noel 3426	. . . . . 6 ⊢ ¬ ⟨𝐴, 𝑥⟩ ∈ ∅
3		df-br 4001	. . . . . 6 ⊢ (𝐴∅𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ ∅)
4	2, 3	mtbir 671	. . . . 5 ⊢ ¬ 𝐴∅𝑥
5	4	nex 1500	. . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥
6		euex 2056	. . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥)
7	5, 6	mto 662	. . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥
8		iotanul 5189	. . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅)
9	7, 8	ax-mp 5	. 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅
10	1, 9	eqtri 2198	1 ⊢ (∅‘𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1353  ∃wex 1492  ∃!weu 2026   ∈ wcel 2148  ∅c0 3422  ⟨cop 3594   class class class wbr 4000  ℩cio 5172  ‘cfv 5212

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220

This theorem is referenced by:  strsl0  12490