Theorem f1imass 7267

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 789	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐴)
2	1	sseld 3937	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐴))
3		simplr 781	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
4	3	sseld 3937	. . . . . . . 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 787	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
6		simpr 490	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7		simp1rl 1257	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	3expa 1136	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
9		f1elima 7266	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
10	5, 6, 8, 9	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
11		simp1rr 1258	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
12	11	3expa 1136	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
13		f1elima 7266	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
14	5, 6, 12, 13	syl3anc 1398	. . . . . . . 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 296	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
16	15	ex 418	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐴 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
17	2, 16	syld 48	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
18	17	pm2.43d 54	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
19	18	ssrdv 3944	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐷)
20	19	ex 418	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) → 𝐶 ⊆ 𝐷))
21		imass2 6106	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
22	20, 21	impbid1 228	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146   ⊆ wss 3906   “ cima 5666  –1-1→wf1 6537  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548

This theorem is used by:  f1imaeq  7268  f1imapss  7269  enfin2i  10320  tsmsf1o  24355  uhgrimisgrgriclem  48755  clnbgrgrimlem  48758