Theorem sspreima 6942

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 6938	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)))
2		df-ss 3909	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	2	biimpi 215	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
4	3	imaeq2d 5968	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ 𝐴))
5	1, 4	sylan9req 2801	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴))
6		df-ss 3909	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴))
7	5, 6	sylibr 233	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∩ cin 3891   ⊆ wss 3892  ^◡ccnv 5589   “ cima 5593  Fun wfun 6426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-fun 6434

This theorem is referenced by:  pwrssmgc  31274  gsumhashmul  31312  elrspunidl  31602  carsggect  32281  eulerpartlemmf  32338  eulerpartlemgf  32342  orvclteinc  32438  cnneiima  46179