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Theorem sspreima 30396
Description: The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
Assertion
Ref Expression
sspreima ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Proof of Theorem sspreima
StepHypRef Expression
1 inpreima 6822 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
2 df-ss 3936 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
32biimpi 219 . . . 4 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
43imaeq2d 5916 . . 3 (𝐴𝐵 → (𝐹 “ (𝐴𝐵)) = (𝐹𝐴))
51, 4sylan9req 2880 . 2 ((Fun 𝐹𝐴𝐵) → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
6 df-ss 3936 . 2 ((𝐹𝐴) ⊆ (𝐹𝐵) ↔ ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
75, 6sylibr 237 1 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  cin 3918  wss 3919  ccnv 5541  cima 5545  Fun wfun 6337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345
This theorem is referenced by:  pwrssmgc  30684  carsggect  31601  eulerpartlemmf  31658  eulerpartlemgf  31662  orvclteinc  31758
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