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Theorem cbvixp 7811
 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,𝑦,𝐴
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 2767 . . . . 5 𝑦(𝑓𝑥) ∈ 𝐵
3 cbvixp.2 . . . . . 6 𝑥𝐶
43nfel2 2767 . . . . 5 𝑥(𝑓𝑦) ∈ 𝐶
5 fveq2 6103 . . . . . 6 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
6 cbvixp.3 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
75, 6eleq12d 2682 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝑓𝑦) ∈ 𝐶))
82, 4, 7cbvral 3143 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)
98anbi2i 726 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶))
109abbii 2726 . 2 {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
11 dfixp 7796 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
12 dfixp 7796 . 2 X𝑦𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
1310, 11, 123eqtr4i 2642 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∀wral 2896   Fn wfn 5799  ‘cfv 5804  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fn 5807  df-fv 5812  df-ixp 7795 This theorem is referenced by:  cbvixpv  7812  mptelixpg  7831  ixpiunwdom  8379  prdsbas3  15964  elptr2  21187  ptunimpt  21208  ptcldmpt  21227  finixpnum  32564  ptrest  32578  hoimbl2  39555  vonhoire  39563  vonn0ioo2  39581  vonn0icc2  39583
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