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Theorem isoeq5 7342
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.
StepHypRef Expression
1 f1oeq3 6837 . . 3 (𝐵 = 𝐶 → (𝐻:𝐴1-1-onto𝐵𝐻:𝐴1-1-onto𝐶))
21anbi1d 631 . 2 (𝐵 = 𝐶 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐴1-1-onto𝐶 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
3 df-isom 6569 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
4 df-isom 6569 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶) ↔ (𝐻:𝐴1-1-onto𝐶 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
52, 3, 43bitr4g 314 1 (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wral 3060   class class class wbr 5142  1-1-ontowf1o 6559  cfv 6560   Isom wiso 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2728  df-ss 3967  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-isom 6569
This theorem is referenced by:  isores3  7356  ordiso  9557  ordtypelem9  9567  ordtypelem10  9568  oiid  9582  iunfictbso  10155  ltweuz  14003  fz1isolem  14501  dvgt0lem2  26043  erdszelem1  35197  erdsze  35208  erdsze2lem1  35209  erdsze2lem2  35210  isoeq145d  43437  alephiso3  43577  fourierdlem50  46176  fourierdlem89  46215  fourierdlem90  46216  fourierdlem91  46217  fourierdlem96  46222  fourierdlem97  46223  fourierdlem98  46224  fourierdlem99  46225  fourierdlem100  46226  fourierdlem108  46234  fourierdlem110  46236  fourierdlem112  46238  fourierdlem113  46239
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