Theorem f1ovscpbl 17579

Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypothesis

Ref	Expression
f1ocpbl.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)

Assertion

Ref	Expression
f1ovscpbl	⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))

Proof of Theorem f1ovscpbl

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)
2		f1of1 6820	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑋 → 𝐹:𝑉–1-1→𝑋)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑋)
4	3	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐹:𝑉–1-1→𝑋)
5		simpr2 1212	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
6		simpr3 1213	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
7		f1fveq 7261	. . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
8	4, 5, 6, 7	syl12anc 849	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
9		oveq2 7419	. . 3 ⊢ (𝐵 = 𝐶 → (𝐴 + 𝐵) = (𝐴 + 𝐶))
10	9	fveq2d 6886	. 2 ⊢ (𝐵 = 𝐶 → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶)))
11	8, 10	biimtrdi 256	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐶) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐴 + 𝐶))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545  df-ov 7414

This theorem is referenced by:  xpsvsca  17630