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Theorem f1imapss 7276
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:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ 𝐶𝐷))

Proof of Theorem f1imapss
StepHypRef Expression
1 f1imass 7274 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))
2 f1imaeq 7275 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))
32notbid 317 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐹𝐶) = (𝐹𝐷) ↔ ¬ 𝐶 = 𝐷))
41, 3anbi12d 630 . 2 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐶) ⊆ (𝐹𝐷) ∧ ¬ (𝐹𝐶) = (𝐹𝐷)) ↔ (𝐶𝐷 ∧ ¬ 𝐶 = 𝐷)))
5 dfpss2 4081 . 2 ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ ((𝐹𝐶) ⊆ (𝐹𝐷) ∧ ¬ (𝐹𝐶) = (𝐹𝐷)))
6 dfpss2 4081 . 2 (𝐶𝐷 ↔ (𝐶𝐷 ∧ ¬ 𝐶 = 𝐷))
74, 5, 63bitr4g 313 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ 𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wss 3944  wpss 3945  cima 5681  1-1wf1 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fv 6557
This theorem is referenced by:  fin4en1  10334
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