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Theorem isoeq4 6470
 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.
StepHypRef Expression
1 f1oeq2 6041 . . 3 (𝐴 = 𝐶 → (𝐻:𝐴1-1-onto𝐵𝐻:𝐶1-1-onto𝐵))
2 raleq 3115 . . . 4 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
32raleqbi1dv 3123 . . 3 (𝐴 = 𝐶 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
41, 3anbi12d 743 . 2 (𝐴 = 𝐶 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
5 df-isom 5813 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 5813 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
74, 5, 63bitr4g 302 1 (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∀wral 2896   class class class wbr 4583  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-isom 5813 This theorem is referenced by:  oieu  8327  oiid  8329  finnisoeu  8819  iunfictbso  8820  fz1isolem  13102  isercolllem3  14245  summolem2a  14293  prodmolem2a  14503  erdszelem1  30427  erdsze  30438  erdsze2lem1  30439  erdsze2lem2  30440  fzisoeu  38455  fourierdlem36  39036  fourierdlem112  39111  fourierdlem113  39112
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