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Theorem funimass2 4977
 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 4701 . 2 (𝐴 ⊆ (𝐹𝐵) → (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)))
2 funimacnv 4975 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
32sseq2d 2973 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) ↔ (𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
4 inss1 3157 . . . . 5 (𝐵 ∩ ran 𝐹) ⊆ 𝐵
5 sstr2 2952 . . . . 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 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∩ cin 2916   ⊆ wss 2917  ◡ccnv 4344  ran crn 4346   “ cima 4348  Fun wfun 4896 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904 This theorem is referenced by:  fvimacnvi  5281
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