Theorem isoeq4 7319

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

Assertion

Ref	Expression
isoeq4	⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))

Proof of Theorem isoeq4

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

Step	Hyp	Ref	Expression
1		f1oeq2 6822	. . 3 ⊢ (𝐴 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐶–1-1-onto→𝐵))
2		raleq 3322	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	raleqbi1dv 3333	. . 3 ⊢ (𝐴 = 𝐶 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
4	1, 3	anbi12d 631	. 2 ⊢ (𝐴 = 𝐶 → ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ↔ (𝐻:𝐶–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))))
5		df-isom 6552	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
6		df-isom 6552	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ (𝐻:𝐶–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
7	4, 5, 6	3bitr4g 313	1 ⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∀wral 3061   class class class wbr 5148  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-ral 3062  df-rex 3071  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-isom 6552

This theorem is referenced by:  oieu  9536  oiid  9538  finnisoeu  10110  iunfictbso  10111  fz1isolem  14426  isercolllem3  15617  summolem2a  15665  prodmolem2a  15882  erdszelem1  34468  erdsze  34479  erdsze2lem1  34480  erdsze2lem2  34481  isoeq145d  42472  fzisoeu  44309  fourierdlem36  45158  fourierdlem112  45233  fourierdlem113  45234