Theorem f1ocpbl 17571

Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
f1ocpbl	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Proof of Theorem f1ocpbl

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)
2	1	f1ocpbllem 17570	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
3		oveq12 7439	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷))
4	3	fveq2d 6910	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)))
5	2, 4	biimtrdi 253	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-f1o 6569  df-fv 6570  df-ov 7433

This theorem is referenced by:  xpsadd  17620  xpsmul  17621  imasmndf1  18801  imasgrpf1  19087  imasrngf1  20195  imasringf1  20344  imasgim  43088