Theorem f1ocpbl 16792

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 16791	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
3		oveq12 7159	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 + 𝐵) = (𝐶 + 𝐷))
4	3	fveq2d 6669	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹‘(𝐴 + 𝐵)) = (𝐹‘(𝐶 + 𝐷)))
5	2, 4	syl6bi 255	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 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-f1o 6357  df-fv 6358  df-ov 7153

This theorem is referenced by:  xpsadd  16841  xpsmul  16842  imasmndf1  17944  imasgrpf1  18210  imasgim  39693