Theorem funimass2 6645

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 6643	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
2	1	sseq2d 3945	. . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) ↔ (𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
3		inss1 4136	. . . 4 ⊢ (𝐵 ∩ ran 𝐹) ⊆ 𝐵
4		sstr2 3920	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵))
5	3, 4	mpi 21	. . 3 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹 “ 𝐴) ⊆ 𝐵)
6	2, 5	biimtrdi 256	. 2 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵))
7		imass2 6067	. 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)))
8	6, 7	impel 515	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3900   ∩ cin 4092  ^◡ccnv 5654  ran crn 5656   “ cima 5658  Fun wfun 6552

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560

This theorem is used by:  fvimacnvi  7074  lmhmlsp  21354  2ndcomap  23807  tgqtop  24061  kqreglem1  24090  fmfnfmlem4  24306  fmucnd  24640  cfilucfil  24908  zarcmplem  34583