ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ixpeq1 GIF version

Theorem ixpeq1 6683
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
ixpeq1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ixpeq1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fneq2 5285 . . . 4 (𝐴 = 𝐵 → (𝑓 Fn 𝐴𝑓 Fn 𝐵))
2 raleq 2665 . . . 4 (𝐴 = 𝐵 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶 ↔ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶))
31, 2anbi12d 470 . . 3 (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)))
43abbidv 2288 . 2 (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)})
5 dfixp 6674 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
6 dfixp 6674 . 2 X𝑥𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥𝐵 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63eqtr4g 2228 1 (𝐴 = 𝐵X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  {cab 2156  wral 2448   Fn wfn 5191  cfv 5196  Xcixp 6672
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-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-fn 5199  df-ixp 6673
This theorem is referenced by:  ixpeq1d  6684
  Copyright terms: Public domain W3C validator