Theorem imadisj 6039

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 5637	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2	1	eqeq1i 2742	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅)
3		dm0rn0 5873	. 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅)
4		dmres 5971	. . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
5		incom 4150	. . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵)
6	4, 5	eqtri 2760	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵)
7	6	eqeq1i 2742	. 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
8	2, 3, 7	3bitr2i 299	1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1542   ∩ cin 3889  ∅c0 4274  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  imadisjlnd  6040  ndmima  6062  fnimadisj  6624  fnimaeq0  6625  fimacnvdisj  6712  frxp2  8087  frxp3  8094  acndom2  9967  isf34lem5  10291  isf34lem7  10292  isf34lem6  10293  limsupgre  15434  isercolllem3  15620  pf1rcl  22324  cnconn  23397  1stcfb  23420  xkohaus  23628  qtopeu  23691  fbasrn  23859  mbflimsup  25643  preiman0  32798  eulerpartlemt  34531  erdszelem5  35393  fnwe2lem2  43497  imadisjld  44605