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Theorem f1ovscpbl 17481
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 6826 . . . . 5 (𝐹:𝑉1-1-onto𝑋𝐹:𝑉1-1𝑋)
31, 2syl 17 . . . 4 (𝜑𝐹:𝑉1-1𝑋)
43adantr 480 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐹:𝑉1-1𝑋)
5 simpr2 1192 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
6 simpr3 1193 . . 3 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
7 f1fveq 7257 . . 3 ((𝐹:𝑉1-1𝑋 ∧ (𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
84, 5, 6, 7syl12anc 834 . 2 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
9 oveq2 7413 . . 3 (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
109fveq2d 6889 . 2 (𝐵 = 𝐶 → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))
118, 10biimtrdi 252 1 ((𝜑 ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐹𝐵) = (𝐹𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  1-1wf1 6534  1-1-ontowf1o 6536  cfv 6537  (class class class)co 7405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545  df-ov 7408
This theorem is referenced by:  xpsvsca  17532
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