Theorem funimass1 4919

Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)

Assertion

Ref	Expression
funimass1	⊢ ((Fun 𝐹 ∧ A ⊆ ran 𝐹) → ((◡𝐹 “ A) ⊆ B → A ⊆ (𝐹 “ B)))

Proof of Theorem funimass1

Step	Hyp	Ref	Expression
1		imass2 4644	. 2 ⊢ ((◡𝐹 “ A) ⊆ B → (𝐹 “ (◡𝐹 “ A)) ⊆ (𝐹 “ B))
2		funimacnv 4918	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ A)) = (A ∩ ran 𝐹))
3		dfss 2926	. . . . . 6 ⊢ (A ⊆ ran 𝐹 ↔ A = (A ∩ ran 𝐹))
4	3	biimpi 113	. . . . 5 ⊢ (A ⊆ ran 𝐹 → A = (A ∩ ran 𝐹))
5	4	eqcomd 2042	. . . 4 ⊢ (A ⊆ ran 𝐹 → (A ∩ ran 𝐹) = A)
6	2, 5	sylan9eq 2089	. . 3 ⊢ ((Fun 𝐹 ∧ A ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ A)) = A)
7	6	sseq1d 2966	. 2 ⊢ ((Fun 𝐹 ∧ A ⊆ ran 𝐹) → ((𝐹 “ (◡𝐹 “ A)) ⊆ (𝐹 “ B) ↔ A ⊆ (𝐹 “ B)))
8	1, 7	syl5ib 143	1 ⊢ ((Fun 𝐹 ∧ A ⊆ ran 𝐹) → ((◡𝐹 “ A) ⊆ B → A ⊆ (𝐹 “ B)))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∩ cin 2910   ⊆ wss 2911  ^◡ccnv 4287  ran crn 4289   “ cima 4291  Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847

This theorem is referenced by: (None)