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Theorem funimass2 5408
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 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)

Proof of Theorem funimass2
StepHypRef Expression
1 imass2 5112 . 2 (𝐴 ⊆ (𝐹𝐵) → (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)))
2 funimacnv 5406 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
32sseq2d 3257 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) ↔ (𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
4 inss1 3427 . . . . 5 (𝐵 ∩ ran 𝐹) ⊆ 𝐵
5 sstr2 3234 . . . . 5 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹𝐴) ⊆ 𝐵))
64, 5mpi 15 . . . 4 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹𝐴) ⊆ 𝐵)
73, 6biimtrdi 163 . . 3 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵))
87imp 124 . 2 ((Fun 𝐹 ∧ (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵))) → (𝐹𝐴) ⊆ 𝐵)
91, 8sylan2 286 1 ((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  cin 3199  wss 3200  ccnv 4724  ran crn 4726  cima 4728  Fun wfun 5320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328
This theorem is referenced by:  fvimacnvi  5761
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