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Theorem fimacnvdisj 5017
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 4299 . . . 4 ran 𝐹 = dom 𝐹
2 frn 4995 . . . . 5 (𝐹:AB → ran 𝐹B)
32adantr 261 . . . 4 ((𝐹:AB (B𝐶) = ∅) → ran 𝐹B)
41, 3syl5eqssr 2984 . . 3 ((𝐹:AB (B𝐶) = ∅) → dom 𝐹B)
5 ssdisj 3271 . . 3 ((dom 𝐹B (B𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
64, 5sylancom 397 . 2 ((𝐹:AB (B𝐶) = ∅) → (dom 𝐹𝐶) = ∅)
7 imadisj 4630 . 2 ((𝐹𝐶) = ∅ ↔ (dom 𝐹𝐶) = ∅)
86, 7sylibr 137 1 ((𝐹:AB (B𝐶) = ∅) → (𝐹𝐶) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cin 2910  wss 2911  c0 3218  ccnv 4287  dom cdm 4288  ran crn 4289  cima 4291  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-f 4849
This theorem is referenced by: (None)
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