Theorem isoeq5 5830

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 5473	. . 3 ⊢ (𝐵 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐴–1-1-onto→𝐶))
2	1	anbi1d 465	. 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 5247	. 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 5247	. 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 223	1 ⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∀wral 2468   class class class wbr 4021  –1-1-onto→wf1o 5237  ‘cfv 5238   Isom wiso 5239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-isom 5247

This theorem is referenced by:  isores3  5840  ordiso  7069  zfz1isolem1  10861  zfz1iso  10862