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Theorem fimacnvdisj 6645
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 5596 . . . 4 ran 𝐹 = dom 𝐹
2 frn 6600 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
32adantr 481 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → ran 𝐹𝐵)
41, 3eqsstrrid 3970 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → dom 𝐹𝐵)
5 ssdisj 4394 . . 3 ((dom 𝐹𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
64, 5sylancom 588 . 2 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
7 imadisj 5982 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
86, 7sylibr 233 1 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  cin 3886  wss 3887  c0 4257  ccnv 5584  dom cdm 5585  ran crn 5586  cima 5588  wf 6423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-xp 5591  df-cnv 5593  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-f 6431
This theorem is referenced by:  vdwmc2  16668  gsumval3a  19492  psrbag0  21258  mbfconstlem  24779  itg1val2  24836  ofpreima2  30989
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