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Theorem f1ovscpbl 17573
Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
f1ocpbl.f (𝜑𝐹:𝑉1-1-onto𝑋)
Assertion
Ref Expression
f1ovscpbl ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))

Proof of Theorem f1ovscpbl
StepHypRef Expression
1 f1ocpbl.f . . . . 5 (𝜑𝐹:𝑉1-1-onto𝑋)
2 f1of1 6848 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
43adantr 480 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐹:𝑉1-1𝑋)
5 simpr2 1194 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
6 simpr3 1195 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
7 f1fveq 7282 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
84, 5, 6, 7syl12anc 837 . 2 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
9 oveq2 7439 . . 3 (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
109fveq2d 6911 . 2 (𝐵 = 𝐶 → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))
118, 10biimtrdi 253 1 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431
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-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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-f1o 6570  df-fv 6571  df-ov 7434
This theorem is referenced by:  xpsvsca  17624
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