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Theorem f1ovscpbl 17572
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 6846 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
43adantr 480 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐹:𝑉1-1𝑋)
5 simpr2 1195 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
6 simpr3 1196 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
7 f1fveq 7283 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
84, 5, 6, 7syl12anc 836 . 2 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
9 oveq2 7440 . . 3 (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
109fveq2d 6909 . 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 1539  wcel 2107  1-1wf1 6557  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-f1o 6567  df-fv 6568  df-ov 7435
This theorem is referenced by:  xpsvsca  17623
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