Theorem isoeq5 7360

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Assertion

Ref	Expression
isoeq5	⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))

Proof of Theorem isoeq5

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

Step	Hyp	Ref	Expression
1		f1oeq3 6844	. . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴⟶1-1onto𝐵 ↔ 𝐻:𝐴⟶1-1onto𝐶))
2	1	anbi1d 643	. 2 ⊢ (𝐵 = 𝐶 → ((𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴⟶1-1onto𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
3		df-isom 6575	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4		df-isom 6575	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶) ↔ (𝐻:𝐴⟶1-1onto𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3080   class class class wbr 5108  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-isom 6575

This theorem is used by:  isores3  7374  ordiso  9547  ordtypelem9  9557  ordtypelem10  9558  oiid  9572  iunfictbso  10232  ltweuz  14144  fz1isolem  14647  dvgt0lem2  26378  erdszelem1  36164  erdsze  36175  erdsze2lem1  36176  erdsze2lem2  36177  isoeq145d  44681  alephiso3  44821  fourierdlem50  47439  fourierdlem89  47478  fourierdlem90  47479  fourierdlem91  47480  fourierdlem96  47485  fourierdlem97  47486  fourierdlem98  47487  fourierdlem99  47488  fourierdlem100  47489  fourierdlem108  47497  fourierdlem110  47499  fourierdlem112  47501  fourierdlem113  47502