Theorem relpf 44940

Description: A relation-preserving function is a function. (Contributed by Eric Schmidt, 11-Oct-2025.)

Assertion

Ref	Expression
relpf	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)

Proof of Theorem relpf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-relp 44933	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
2	1	simplbi 497	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)

Syntax hints:   → wi 4  ∀wral 3044   class class class wbr 5107  ⟶wf 6507  ‘cfv 6511   RelPres wrelp 44932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-relp 44933

This theorem is referenced by:  relpmin  44942  relpfrlem  44943  relpfr  44944