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Theorem f1ovscpbl 16107
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 6093 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
43adantr 481 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐹:𝑉1-1𝑋)
5 simpr2 1066 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
6 simpr3 1067 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
7 f1fveq 6473 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
84, 5, 6, 7syl12anc 1321 . 2 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
9 oveq2 6612 . . 3 (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
109fveq2d 6152 . 2 (𝐵 = 𝐶 → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))
118, 10syl6bi 243 1 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-f1o 5854  df-fv 5855  df-ov 6607
This theorem is referenced by:  xpsvsca  16160
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