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Theorem fimacnvdisj 6775
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 5689 . . . 4 ran 𝐹 = dom 𝐹
2 frn 6730 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
32adantr 479 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → ran 𝐹𝐵)
41, 3eqsstrrid 4026 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → dom 𝐹𝐵)
5 ssdisj 4461 . . 3 ((dom 𝐹𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
64, 5sylancom 586 . 2 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
7 imadisj 6084 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
86, 7sylibr 233 1 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  cin 3943  wss 3944  c0 4322  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  wf 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-f 6553
This theorem is referenced by:  vdwmc2  16951  gsumval3a  19870  psrbag0  22028  mbfconstlem  25600  itg1val2  25657  ofpreima2  32533
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