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Theorem f1olecpbl 16010
Description: An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypothesis
Ref Expression
f1ocpbl.f (𝜑𝐹:𝑉1-1-onto𝑋)
Assertion
Ref Expression
f1olecpbl ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐴𝑁𝐵𝐶𝑁𝐷)))

Proof of Theorem f1olecpbl
StepHypRef Expression
1 f1ocpbl.f . . 3 (𝜑𝐹:𝑉1-1-onto𝑋)
21f1ocpbllem 16007 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
3 breq12 4588 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝑁𝐵𝐶𝑁𝐷))
42, 3syl6bi 242 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐴𝑁𝐵𝐶𝑁𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  1-1-ontowf1o 5803  cfv 5804
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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812
This theorem is referenced by:  xpsle  16064
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