Theorem sspreima 7006

Description: The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)

Assertion

Ref	Expression
sspreima	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵))

Proof of Theorem sspreima

Step	Hyp	Ref	Expression
1		inpreima 7002	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)))
2		dfss2 3923	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	2	biimpi 216	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
4	3	imaeq2d 6015	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ 𝐴))
5	1, 4	sylan9req 2785	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴))
6		dfss2 3923	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴))
7	5, 6	sylibr 234	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3904   ⊆ wss 3905  ^◡ccnv 5622   “ cima 5626  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488

This theorem is referenced by:  pwrssmgc  32955  gsumhashmul  33027  elrspunidl  33375  carsggect  34285  eulerpartlemmf  34342  eulerpartlemgf  34346  orvclteinc  34443  cnneiima  48902