Theorem strlemor0OLD 16015

Description: Structure definition utility lemma. To prove that an explicit function is a function using O(n) steps, exploit the order properties of the index set. Zero-pair case. Obsolete as of 26-Nov-2021. Theorems strlemor0OLD 16015, strlemor1OLD 16016, strlemor2OLD 16017, strlemor3OLD 16018 were replaced by strleun 16019, strle1 16020, strle2 16021, strle3 16022 following the introduction df-struct 15906. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
strlemor0OLD	⊢ (Fun ◡◡∅ ∧ dom ∅ ⊆ (1...0))

Proof of Theorem strlemor0OLD

Step	Hyp	Ref	Expression
1		fun0 5992	. . 3 ⊢ Fun ∅
2		funcnvcnv 5994	. . 3 ⊢ (Fun ∅ → Fun ◡◡∅)
3	1, 2	ax-mp 5	. 2 ⊢ Fun ◡◡∅
4		dm0 5371	. . 3 ⊢ dom ∅ = ∅
5		0ss 4005	. . 3 ⊢ ∅ ⊆ (1...0)
6	4, 5	eqsstri 3668	. 2 ⊢ dom ∅ ⊆ (1...0)
7	3, 6	pm3.2i 470	1 ⊢ (Fun ◡◡∅ ∧ dom ∅ ⊆ (1...0))

Syntax hints:   ∧ wa 383   ⊆ wss 3607  ∅c0 3948  ^◡ccnv 5142  dom cdm 5143  Fun wfun 5920  (class class class)co 6690  0cc0 9974  1c1 9975  ...cfz 12364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-fun 5928

This theorem is referenced by: (None)