Theorem funimass1 6631

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 6102	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ 𝐵 → (𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵))
2		funimacnv 6630	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
3		dfss 3967	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ 𝐴 = (𝐴 ∩ ran 𝐹))
4	3	biimpi 215	. . . . 5 ⊢ (𝐴 ⊆ ran 𝐹 → 𝐴 = (𝐴 ∩ ran 𝐹))
5	4	eqcomd 2739	. . . 4 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
6	2, 5	sylan9eq 2793	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
7	6	sseq1d 4014	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵) ↔ 𝐴 ⊆ (𝐹 “ 𝐵)))
8	1, 7	imbitrid 243	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) ⊆ 𝐵 → 𝐴 ⊆ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∩ cin 3948   ⊆ wss 3949  ^◡ccnv 5676  ran crn 5678   “ cima 5680  Fun wfun 6538

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546

This theorem is referenced by:  kqnrmlem1  23247  hmeontr  23273  nrmhmph  23298  cnheiborlem  24470