Theorem sspreima 7001

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 6997	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)))
2		dfss2 3920	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
3	2	biimpi 216	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
4	3	imaeq2d 6009	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ 𝐴))
5	1, 4	sylan9req 2787	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴))
6		dfss2 3920	. 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 1541   ∩ cin 3901   ⊆ wss 3902  ^◡ccnv 5615   “ cima 5619  Fun wfun 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483

This theorem is referenced by:  pwrssmgc  32976  gsumhashmul  33036  elrspunidl  33388  carsggect  34326  eulerpartlemmf  34383  eulerpartlemgf  34387  orvclteinc  34484  cnneiima  48947