Theorem f1imapss 7243

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

Assertion

Ref	Expression
f1imapss	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷))

Proof of Theorem f1imapss

Step	Hyp	Ref	Expression
1		f1imass 7241	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imaeq 7242	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))
3	2	notbid 318	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ¬ 𝐶 = 𝐷))
4	1, 3	anbi12d 632	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ ¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷)) ↔ (𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷)))
5		dfpss2 4053	. 2 ⊢ ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ ¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷)))
6		dfpss2 4053	. 2 ⊢ (𝐶 ⊊ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷))
7	4, 5, 6	3bitr4g 314	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ⊆ wss 3916   ⊊ wpss 3917   “ cima 5643  –1-1→wf1 6510

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fv 6521

This theorem is referenced by:  fin4en1  10268