Theorem funimass2 5410

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Assertion

Ref	Expression
funimass2	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)

Proof of Theorem funimass2

Step	Hyp	Ref	Expression
1		imass2 5114	. 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)))
2		funimacnv 5408	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
3	2	sseq2d 3256	. . . 4 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) ↔ (𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
4		inss1 3426	. . . . 5 ⊢ (𝐵 ∩ ran 𝐹) ⊆ 𝐵
5		sstr2 3233	. . . . 5 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵))
6	4, 5	mpi 15	. . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹 “ 𝐴) ⊆ 𝐵)
7	3, 6	biimtrdi 163	. . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵))
8	7	imp 124	. 2 ⊢ ((Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵))) → (𝐹 “ 𝐴) ⊆ 𝐵)
9	1, 8	sylan2 286	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∩ cin 3198   ⊆ wss 3199  ^◡ccnv 4726  ran crn 4728   “ cima 4730  Fun wfun 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-fun 5330

This theorem is referenced by:  fvimacnvi  5764