Theorem isoeq5 7320

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 6811	. . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐴–1-1-onto→𝐶))
2	1	anbi1d 642	. 2 ⊢ (𝐵 = 𝐶 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐴–1-1-onto→𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
3		df-isom 6546	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4		df-isom 6546	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶) ↔ (𝐻:𝐴–1-1-onto→𝐶 ∧ ∀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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∀wral 3085   class class class wbr 5113  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ss 3930  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-isom 6546

This theorem is referenced by:  isores3  7334  ordiso  9477  ordtypelem9  9487  ordtypelem10  9488  oiid  9502  iunfictbso  10097  ltweuz  13996  fz1isolem  14497  dvgt0lem2  26130  erdszelem1  35581  erdsze  35592  erdsze2lem1  35593  erdsze2lem2  35594  isoeq145d  44036  alephiso3  44176  fourierdlem50  46761  fourierdlem89  46800  fourierdlem90  46801  fourierdlem91  46802  fourierdlem96  46807  fourierdlem97  46808  fourierdlem98  46809  fourierdlem99  46810  fourierdlem100  46811  fourierdlem108  46819  fourierdlem110  46821  fourierdlem112  46823  fourierdlem113  46824