Theorem f1imass 5914

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 3226	. . . . . 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 3226	. . . . . . . 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 1088	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
8	7	3expa 1229	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐶 ⊆ 𝐴)
9		f1elima 5913	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
10	5, 6, 8, 9	syl3anc 1273	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐶) ↔ 𝑎 ∈ 𝐶))
11		simp1rr 1089	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
12	11	3expa 1229	. . . . . . . . 9 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝐷 ⊆ 𝐴)
13		f1elima 5913	. . . . . . . . 9 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑎 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((𝐹‘𝑎) ∈ (𝐹 “ 𝐷) ↔ 𝑎 ∈ 𝐷))
14	5, 6, 12, 13	syl3anc 1273	. . . . . . . 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 3233	. . 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 5112	. 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 2202   ⊆ wss 3200   “ cima 4728  –1-1→wf1 5323  ‘cfv 5326

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334

This theorem is referenced by:  f1imaeq  5915