Theorem f1olecpbl 16010

Description: An injection is compatible with any relations 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
f1olecpbl	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))

Proof of Theorem f1olecpbl

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)
2	1	f1ocpbllem 16007	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
3		breq12 4588	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷))
4	2, 3	syl6bi 242	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) → (𝐴𝑁𝐵 ↔ 𝐶𝑁𝐷)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  –1-1-onto→wf1o 5803  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-f1o 5811  df-fv 5812

This theorem is referenced by:  xpsle  16064