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Theorem isoeq4 7339
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 6837 . . 3 (𝐴 = 𝐶 → (𝐻:𝐴1-1-onto𝐵𝐻:𝐶1-1-onto𝐵))
2 raleq 3320 . . . 4 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
32raleqbi1dv 3335 . . 3 (𝐴 = 𝐶 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
41, 3anbi12d 632 . 2 (𝐴 = 𝐶 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
5 df-isom 6571 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 6571 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
74, 5, 63bitr4g 314 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 206  wa 395   = wceq 1536  wral 3058   class class class wbr 5147  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-ral 3059  df-rex 3068  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-isom 6571
This theorem is referenced by:  oieu  9576  oiid  9578  finnisoeu  10150  iunfictbso  10151  fz1isolem  14496  isercolllem3  15699  summolem2a  15747  prodmolem2a  15966  erdszelem1  35175  erdsze  35186  erdsze2lem1  35187  erdsze2lem2  35188  isoeq145d  43408  fzisoeu  45250  fourierdlem36  46098  fourierdlem112  46173  fourierdlem113  46174
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