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Theorem funimass1 6624
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 6095 . 2 ((𝐹𝐴) ⊆ 𝐵 → (𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵))
2 funimacnv 6623 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
3 dfss 3961 . . . . . 6 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
43biimpi 215 . . . . 5 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
54eqcomd 2732 . . . 4 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
62, 5sylan9eq 2786 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐹 “ (𝐹𝐴)) = 𝐴)
76sseq1d 4008 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵) ↔ 𝐴 ⊆ (𝐹𝐵)))
81, 7imbitrid 243 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  cin 3942  wss 3943  ccnv 5668  ran crn 5670  cima 5672  Fun wfun 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6539
This theorem is referenced by:  kqnrmlem1  23602  hmeontr  23628  nrmhmph  23653  cnheiborlem  24835
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