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Theorem isoeq4 5783
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 5432 . . 3 (𝐴 = 𝐶 → (𝐻:𝐴1-1-onto𝐵𝐻:𝐶1-1-onto𝐵))
2 raleq 2665 . . . 4 (𝐴 = 𝐶 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
32raleqbi1dv 2673 . . 3 (𝐴 = 𝐶 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
41, 3anbi12d 470 . 2 (𝐴 = 𝐶 → ((𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))))
5 df-isom 5207 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
6 df-isom 5207 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ (𝐻:𝐶1-1-onto𝐵 ∧ ∀𝑥𝐶𝑦𝐶 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
74, 5, 63bitr4g 222 1 (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wral 2448   class class class wbr 3989  1-1-ontowf1o 5197  cfv 5198   Isom wiso 5199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-isom 5207
This theorem is referenced by:  zfz1isolem1  10775  zfz1iso  10776  summodclem2a  11344  prodmodclem2a  11539
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