Theorem f1imapss 7200

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 7198	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imaeq 7199	. . . 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 4038	. 2 ⊢ ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ ¬ (𝐹 “ 𝐶) = (𝐹 “ 𝐷)))
6		dfpss2 4038	. 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 1541   ⊆ wss 3902   ⊊ wpss 3903   “ cima 5619  –1-1→wf1 6478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fv 6489

This theorem is referenced by:  fin4en1  10200