Theorem funimass1 6204

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 5742	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ 𝐵 → (𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵))
2		funimacnv 6203	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
3		dfss 3813	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ 𝐴 = (𝐴 ∩ ran 𝐹))
4	3	biimpi 208	. . . . 5 ⊢ (𝐴 ⊆ ran 𝐹 → 𝐴 = (𝐴 ∩ ran 𝐹))
5	4	eqcomd 2831	. . . 4 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
6	2, 5	sylan9eq 2881	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
7	6	sseq1d 3857	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵) ↔ 𝐴 ⊆ (𝐹 “ 𝐵)))
8	1, 7	syl5ib 236	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) ⊆ 𝐵 → 𝐴 ⊆ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∩ cin 3797   ⊆ wss 3798  ^◡ccnv 5341  ran crn 5343   “ cima 5345  Fun wfun 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-fun 6125

This theorem is referenced by:  kqnrmlem1  21917  hmeontr  21943  nrmhmph  21968  cnheiborlem  23123