Theorem f1imass 5947

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 537	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐴)
2	1	sseld 3237	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐴))
3		simplr 529	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
4	3	sseld 3237	. . . . . . . 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 535	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
6		simpr 110	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7		simp1rl 1089	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	3expa 1230	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
9		f1elima 5946	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
10	5, 6, 8, 9	syl3anc 1274	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
11		simp1rr 1090	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
12	11	3expa 1230	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
13		f1elima 5946	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
14	5, 6, 12, 13	syl3anc 1274	. . . . . . . 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 202	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
16	15	ex 115	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐴 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
17	2, 16	syld 45	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷)))
18	17	pm2.43d 50	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
19	18	ssrdv 3244	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐷)
20	19	ex 115	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) → 𝐶 ⊆ 𝐷))
21		imass2 5138	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
22	20, 21	impbid1 142	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2203   ⊆ wss 3211   “ cima 4752  –1-1→wf1 5349  ‘cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fv 5360

This theorem is referenced by:  f1imaeq  5948