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Theorem fimacnvdisj 6223
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 5260 . . . 4 ran 𝐹 = dom 𝐹
2 frn 6193 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
32adantr 466 . . . 4 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → ran 𝐹𝐵)
41, 3syl5eqssr 3799 . . 3 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → dom 𝐹𝐵)
5 ssdisj 4170 . . 3 ((dom 𝐹𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
64, 5sylancom 576 . 2 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
7 imadisj 5625 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
86, 7sylibr 224 1 ((𝐹:𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  cin 3722  wss 3723  c0 4063  ccnv 5248  dom cdm 5249  ran crn 5250  cima 5252  wf 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-f 6035
This theorem is referenced by:  vdwmc2  15890  gsumval3a  18511  psrbag0  19709  mbfconstlem  23615  itg1val2  23671  ofpreima2  29806
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