Theorem imadisj 5941

Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
imadisj	⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)

Proof of Theorem imadisj

Step	Hyp	Ref	Expression
1		df-ima 5561	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2	1	eqeq1i 2824	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅)
3		dm0rn0 5788	. 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅)
4		dmres 5868	. . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
5		incom 4176	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
6	4, 5	eqtri 2842	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵)
7	6	eqeq1i 2824	. 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
8	2, 3, 7	3bitr2i 301	1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)

Syntax hints:   ↔ wb 208   = wceq 1531   ∩ cin 3933  ∅c0 4289  dom cdm 5548  ran crn 5549   ↾ cres 5550   “ cima 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561

This theorem is referenced by:  ndmima  5959  fnimadisj  6473  fnimaeq0  6474  fimacnvdisj  6550  acndom2  9472  isf34lem5  9792  isf34lem7  9793  isf34lem6  9794  limsupgre  14830  isercolllem3  15015  pf1rcl  20504  cnconn  22022  1stcfb  22045  xkohaus  22253  qtopeu  22316  fbasrn  22484  mbflimsup  24259  eulerpartlemt  31622  erdszelem5  32435  fnwe2lem2  39641  imadisjld  40500  imadisjlnd  40501