Theorem funimass1 5435

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 5140	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ 𝐵 → (𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵))
2		funimacnv 5434	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
3		dfss 3227	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ 𝐴 = (𝐴 ∩ ran 𝐹))
4	3	biimpi 120	. . . . 5 ⊢ (𝐴 ⊆ ran 𝐹 → 𝐴 = (𝐴 ∩ ran 𝐹))
5	4	eqcomd 2240	. . . 4 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
6	2, 5	sylan9eq 2287	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
7	6	sseq1d 3269	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵) ↔ 𝐴 ⊆ (𝐹 “ 𝐵)))
8	1, 7	imbitrid 154	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) ⊆ 𝐵 → 𝐴 ⊆ (𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∩ cin 3212   ⊆ wss 3213  ^◡ccnv 4750  ran crn 4752   “ cima 4754  Fun wfun 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356

This theorem is referenced by:  hmeontr  15227