Theorem f1imass 7257

Description: Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Assertion

Ref	Expression
f1imass	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))

Proof of Theorem f1imass

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplrl 776	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐴)
2	1	sseld 3957	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐴))
3		simplr 768	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
4	3	sseld 3957	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) → (𝐹‘𝑎) ∈ (𝐹 “ 𝐷)))
5		simplll 774	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
6		simpr 484	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7		simp1rl 1239	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	3expa 1118	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
9		f1elima 7256	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
10	5, 6, 8, 9	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
11		simp1rr 1240	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
12	11	3expa 1118	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
13		f1elima 7256	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
14	5, 6, 12, 13	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
15	4, 10, 14	3imtr3d 293	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
16	15	ex 412	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐴 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
17	2, 16	syld 47	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
18	17	pm2.43d 53	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
19	18	ssrdv 3964	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐷)
20	19	ex 412	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) → 𝐶 ⊆ 𝐷))
21		imass2 6089	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
22	20, 21	impbid1 225	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3926   “ cima 5657  –1-1→wf1 6528  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fv 6539

This theorem is referenced by:  f1imaeq  7258  f1imapss  7259  enfin2i  10335  tsmsf1o  24083  uhgrimisgrgriclem  47943  clnbgrgrimlem  47946