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Theorem sspreima 29787
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 6485 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
2 df-ss 3737 . . . . 5 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
32biimpi 206 . . . 4 (𝐴𝐵 → (𝐴𝐵) = 𝐴)
43imaeq2d 5607 . . 3 (𝐴𝐵 → (𝐹 “ (𝐴𝐵)) = (𝐹𝐴))
51, 4sylan9req 2826 . 2 ((Fun 𝐹𝐴𝐵) → ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
6 df-ss 3737 . 2 ((𝐹𝐴) ⊆ (𝐹𝐵) ↔ ((𝐹𝐴) ∩ (𝐹𝐵)) = (𝐹𝐴))
75, 6sylibr 224 1 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  cin 3722  wss 3723  ccnv 5248  cima 5252  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-fun 6033
This theorem is referenced by:  carsggect  30720  eulerpartlemmf  30777  eulerpartlemgf  30781  orvclteinc  30877
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