Theorem strlemor0OLD 15968

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 15968, strlemor1OLD 15969, strlemor2OLD 15970, strlemor3OLD 15971 were replaced by strleun 15972, strle1 15973, strle2 15974, strle3 15975 following the introduction df-struct 15859. (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 5954	. . 3 ⊢ Fun ∅
2		funcnvcnv 5956	. . 3 ⊢ (Fun ∅ → Fun ◡◡∅)
3	1, 2	ax-mp 5	. 2 ⊢ Fun ◡◡∅
4		dm0 5339	. . 3 ⊢ dom ∅ = ∅
5		0ss 3972	. . 3 ⊢ ∅ ⊆ (1...0)
6	4, 5	eqsstri 3635	. 2 ⊢ dom ∅ ⊆ (1...0)
7	3, 6	pm3.2i 471	1 ⊢ (Fun ◡◡∅ ∧ dom ∅ ⊆ (1...0))

Syntax hints:   ∧ wa 384   ⊆ wss 3574  ∅c0 3915  ^◡ccnv 5113  dom cdm 5114  Fun wfun 5882  (class class class)co 6650  0cc0 9936  1c1 9937  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890

This theorem is referenced by: (None)