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Theorem f1olecpbl 16800
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 16797 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
3 breq12 5057 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴𝑁𝐵𝐶𝑁𝐷))
42, 3syl6bi 256 1 ((𝜑 ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (((𝐹𝐴) = (𝐹𝐶) ∧ (𝐹𝐵) = (𝐹𝐷)) → (𝐴𝑁𝐵𝐶𝑁𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5052  1-1-ontowf1o 6342  cfv 6343
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-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  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-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-f1o 6350  df-fv 6351
This theorem is referenced by:  xpsle  16852
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