Theorem funimass1 5293

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 5004	. 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ 𝐵 → (𝐹 “ (◡𝐹 “ 𝐴)) ⊆ (𝐹 “ 𝐵))
2		funimacnv 5292	. . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
3		dfss 3143	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 ↔ 𝐴 = (𝐴 ∩ ran 𝐹))
4	3	biimpi 120	. . . . 5 ⊢ (𝐴 ⊆ ran 𝐹 → 𝐴 = (𝐴 ∩ ran 𝐹))
5	4	eqcomd 2183	. . . 4 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
6	2, 5	sylan9eq 2230	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐹 “ (◡𝐹 “ 𝐴)) = 𝐴)
7	6	sseq1d 3184	. 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 1353   ∩ cin 3128   ⊆ wss 3129  ^◡ccnv 4625  ran crn 4627   “ cima 4629  Fun wfun 5210

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-fun 5218

This theorem is referenced by:  hmeontr  13749