Theorem f1imapss 7265

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 7263	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imaeq 7264	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))
3	2	notbid 321	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ¬ 𝐶 = 𝐷))
4	1, 3	anbi12d 643	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ ¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷)) ↔ (𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷)))
5		dfpss2 4050	. 2 ⊢ ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ ¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷)))
6		dfpss2 4050	. 2 ⊢ (𝐶 ⊊ 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ ¬ 𝐶 = 𝐷))
7	4, 5, 6	3bitr4g 317	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ⊆ wss 3913   ⊊ wpss 3914   “ cima 5665  –1-1→wf1 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fv 6545

This theorem is referenced by:  fin4en1  10292