Theorem isoeq4 6894

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 6431	. . 3 ⊢ (𝐴 = 𝐶 → (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻:𝐶–1-1-onto→𝐵))
2		raleq 3338	. . . 4 ⊢ (𝐴 = 𝐶 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
3	2	raleqbi1dv 3336	. . 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 622	. 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 6194	. 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 6194	. 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 306	1 ⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508  ∀wral 3081   class class class wbr 4925  –1-1-onto→wf1o 6184  ‘cfv 6185   Isom wiso 6186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2764  df-clel 2839  df-ral 3086  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-isom 6194

This theorem is referenced by:  oieu  8796  oiid  8798  finnisoeu  9331  iunfictbso  9332  fz1isolem  13630  isercolllem3  14882  summolem2a  14930  prodmolem2a  15146  erdszelem1  32060  erdsze  32071  erdsze2lem1  32072  erdsze2lem2  32073  fzisoeu  41028  fourierdlem36  41891  fourierdlem112  41966  fourierdlem113  41967