Theorem funimass1 5374

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 5080	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ 𝐵 → (𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵))
2		funimacnv 5373	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
3		dfss 3191	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ 𝐴 = (𝐴 ∩ ran 𝐹))
4	3	biimpi 120	. . . . 5 ⊢ (𝐴 ⊆ ran 𝐹 → 𝐴 = (𝐴 ∩ ran 𝐹))
5	4	eqcomd 2215	. . . 4 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
6	2, 5	sylan9eq 2262	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
7	6	sseq1d 3233	. 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 1375   ∩ cin 3176   ⊆ wss 3177  ^◡ccnv 4695  ran crn 4697   “ cima 4699  Fun wfun 5288

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-fun 5296

This theorem is referenced by:  hmeontr  14952