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Theorem funimass2 4920
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass2 ((Fun 𝐹 A ⊆ (𝐹B)) → (𝐹A) ⊆ B)

Proof of Theorem funimass2
StepHypRef Expression
1 imass2 4644 . 2 (A ⊆ (𝐹B) → (𝐹A) ⊆ (𝐹 “ (𝐹B)))
2 funimacnv 4918 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹B)) = (B ∩ ran 𝐹))
32sseq2d 2967 . . . 4 (Fun 𝐹 → ((𝐹A) ⊆ (𝐹 “ (𝐹B)) ↔ (𝐹A) ⊆ (B ∩ ran 𝐹)))
4 inss1 3151 . . . . 5 (B ∩ ran 𝐹) ⊆ B
5 sstr2 2946 . . . . 5 ((𝐹A) ⊆ (B ∩ ran 𝐹) → ((B ∩ ran 𝐹) ⊆ B → (𝐹A) ⊆ B))
64, 5mpi 15 . . . 4 ((𝐹A) ⊆ (B ∩ ran 𝐹) → (𝐹A) ⊆ B)
73, 6syl6bi 152 . . 3 (Fun 𝐹 → ((𝐹A) ⊆ (𝐹 “ (𝐹B)) → (𝐹A) ⊆ B))
87imp 115 . 2 ((Fun 𝐹 (𝐹A) ⊆ (𝐹 “ (𝐹B))) → (𝐹A) ⊆ B)
91, 8sylan2 270 1 ((Fun 𝐹 A ⊆ (𝐹B)) → (𝐹A) ⊆ B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  cin 2910  wss 2911  ccnv 4287  ran crn 4289  cima 4291  Fun wfun 4839
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-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-bnd 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-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-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by:  fvimacnvi  5224
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