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Theorem f1imaeq 6423
 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 6422 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))
2 f1imass 6422 . . . 4 ((𝐹:𝐴1-1𝐵 ∧ (𝐷𝐴𝐶𝐴)) → ((𝐹𝐷) ⊆ (𝐹𝐶) ↔ 𝐷𝐶))
32ancom2s 840 . . 3 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐷) ⊆ (𝐹𝐶) ↔ 𝐷𝐶))
41, 3anbi12d 743 . 2 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → (((𝐹𝐶) ⊆ (𝐹𝐷) ∧ (𝐹𝐷) ⊆ (𝐹𝐶)) ↔ (𝐶𝐷𝐷𝐶)))
5 eqss 3583 . 2 ((𝐹𝐶) = (𝐹𝐷) ↔ ((𝐹𝐶) ⊆ (𝐹𝐷) ∧ (𝐹𝐷) ⊆ (𝐹𝐶)))
6 eqss 3583 . 2 (𝐶 = 𝐷 ↔ (𝐶𝐷𝐷𝐶))
74, 5, 63bitr4g 302 1 ((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540   “ cima 5041  –1-1→wf1 5801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812 This theorem is referenced by:  f1imapss  6424  dfac12lem2  8849  hmeoimaf1o  21383  imasf1oxms  22104
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