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Theorem f1imapss 7015
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 7013 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))
2 f1imaeq 7014 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))
32notbid 319 . . 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 4059 . 2 ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ ((𝐹𝐶) ⊆ (𝐹𝐷) ∧ ¬ (𝐹𝐶) = (𝐹𝐷)))
6 dfpss2 4059 . 2 (𝐶𝐷 ↔ (𝐶𝐷 ∧ ¬ 𝐶 = 𝐷))
74, 5, 63bitr4g 315 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ 𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wss 3933  wpss 3934  cima 5551  1-1wf1 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356
This theorem is referenced by:  fin4en1  9719
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