Theorem f1imass 7208

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 782	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐴)
2	1	sseld 3914	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐴))
3		simplr 774	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
4	3	sseld 3914	. . . . . . . 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 780	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
6		simpr 485	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7		simp1rl 1245	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	3expa 1124	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
9		f1elima 7207	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
10	5, 6, 8, 9	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
11		simp1rr 1246	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
12	11	3expa 1124	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
13		f1elima 7207	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
14	5, 6, 12, 13	syl3anc 1379	. . . . . . . 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 294	. . . . . . 7 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝐷))
16	15	ex 413	. . . . . 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 3921	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) → 𝐶 ⊆ 𝐷)
20	19	ex 413	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) → 𝐶 ⊆ 𝐷))
21		imass2 6054	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷))
22	20, 21	impbid1 226	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119   ⊆ wss 3883   “ cima 5621  –1-1→wf1 6482  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fv 6493

This theorem is referenced by:  f1imaeq  7209  f1imapss  7210  enfin2i  10234  tsmsf1o  24128  uhgrimisgrgriclem  48421  clnbgrgrimlem  48424