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Theorem f1imapss 7016
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 7014 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))
2 f1imaeq 7015 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))
32notbid 321 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (¬ (𝐹𝐶) = (𝐹𝐷) ↔ ¬ 𝐶 = 𝐷))
41, 3anbi12d 633 . 2 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐶) ⊆ (𝐹𝐷) ∧ ¬ (𝐹𝐶) = (𝐹𝐷)) ↔ (𝐶𝐷 ∧ ¬ 𝐶 = 𝐷)))
5 dfpss2 4048 . 2 ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ ((𝐹𝐶) ⊆ (𝐹𝐷) ∧ ¬ (𝐹𝐶) = (𝐹𝐷)))
6 dfpss2 4048 . 2 (𝐶𝐷 ↔ (𝐶𝐷 ∧ ¬ 𝐶 = 𝐷))
74, 5, 63bitr4g 317 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ 𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wss 3919  wpss 3920  cima 5545  1-1wf1 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fv 6351
This theorem is referenced by:  fin4en1  9729
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