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Theorem f1imaeq 7285
Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Assertion
Ref Expression
f1imaeq ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))

Proof of Theorem f1imaeq
StepHypRef Expression
1 f1imass 7284 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))
2 f1imass 7284 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝐷𝐴𝐶𝐴)) → ((𝐹𝐷) ⊆ (𝐹𝐶) ↔ 𝐷𝐶))
32ancom2s 650 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐷) ⊆ (𝐹𝐶) ↔ 𝐷𝐶))
41, 3anbi12d 632 . 2 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐶) ⊆ (𝐹𝐷) ∧ (𝐹𝐷) ⊆ (𝐹𝐶)) ↔ (𝐶𝐷𝐷𝐶)))
5 eqss 4011 . 2 ((𝐹𝐶) = (𝐹𝐷) ↔ ((𝐹𝐶) ⊆ (𝐹𝐷) ∧ (𝐹𝐷) ⊆ (𝐹𝐶)))
6 eqss 4011 . 2 (𝐶 = 𝐷 ↔ (𝐶𝐷𝐷𝐶))
74, 5, 63bitr4g 314 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wss 3963  cima 5692  1-1wf1 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571
This theorem is referenced by:  f1imapss  7286  dfac12lem2  10183  hmeoimaf1o  23794  imasf1oxms  24518  isuspgrim0  47810  isuspgrimlem  47812  grimedg  47841
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