Theorem relpf 45700

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 45693	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
2	1	simplbi 502	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4  ∀wral 3082   class class class wbr 5114  ⟶wf 6539  ‘cfv 6543   RelPres wrelp 45692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-relp 45693

This theorem is used by:  relpmin  45702  relpfrlem  45703  relpfr  45704