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Theorem cbvixp 8953
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
Hypotheses
Ref Expression
cbvixp.1 𝑦𝐵
cbvixp.2 𝑥𝐶
cbvixp.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvixp X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvixp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cbvixp.1 . . . . . 6 𝑦𝐵
21nfel2 2922 . . . . 5 𝑦(𝑓𝑥) ∈ 𝐵
3 cbvixp.2 . . . . . 6 𝑥𝐶
43nfel2 2922 . . . . 5 𝑥(𝑓𝑦) ∈ 𝐶
5 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
6 cbvixp.3 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
75, 6eleq12d 2833 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝑓𝑦) ∈ 𝐶))
82, 4, 7cbvralw 3304 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)
98anbi2i 623 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶))
109abbii 2807 . 2 {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
11 dfixp 8938 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
12 dfixp 8938 . 2 X𝑦𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
1310, 11, 123eqtr4i 2773 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wnfc 2888  wral 3059   Fn wfn 6558  cfv 6563  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fn 6566  df-fv 6571  df-ixp 8937
This theorem is referenced by:  mptelixpg  8974  ixpiunwdom  9628  prdsbas3  17528  elptr2  23598  ptunimpt  23619  ptcldmpt  23638  finixpnum  37592  ptrest  37606  hoimbl2  46621  vonhoire  46628  vonn0ioo2  46646  vonn0icc2  46648
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