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Theorem fimacnvdisj 6785
Description: The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
fimacnvdisj ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)

Proof of Theorem fimacnvdisj
StepHypRef Expression
1 df-rn 5695 . . . 4 ran 𝐹 = dom 𝐹
2 frn 6742 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
32adantr 480 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → ran 𝐹𝐵)
41, 3eqsstrrid 4022 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → dom 𝐹𝐵)
5 ssdisj 4459 . . 3 ((dom 𝐹𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
64, 5sylancom 588 . 2 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
7 imadisj 6097 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
86, 7sylibr 234 1 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  cin 3949  wss 3950  c0 4332  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-f 6564
This theorem is referenced by:  vdwmc2  17018  gsumval3a  19922  psrbag0  22087  mbfconstlem  25663  itg1val2  25720  ofpreima2  32677
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