Theorem funimass2 6517

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

Assertion

Ref	Expression
funimass2	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)

Proof of Theorem funimass2

Step	Hyp	Ref	Expression
1		funimacnv 6515	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
2	1	sseq2d 3953	. . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) ↔ (𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
3		inss1 4162	. . . 4 ⊢ (𝐵 ∩ ran 𝐹) ⊆ 𝐵
4		sstr2 3928	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵))
5	3, 4	mpi 20	. . 3 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹 “ 𝐴) ⊆ 𝐵)
6	2, 5	syl6bi 252	. 2 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵))
7		imass2 6010	. 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)))
8	6, 7	impel 506	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∩ cin 3886   ⊆ wss 3887  ^◡ccnv 5588  ran crn 5590   “ cima 5592  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435

This theorem is referenced by:  fvimacnvi  6929  lmhmlsp  20311  2ndcomap  22609  tgqtop  22863  kqreglem1  22892  fmfnfmlem4  23108  fmucnd  23444  cfilucfil  23715  zarcmplem  31831