MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvixp Structured version   Visualization version   GIF version

Theorem cbvixp 8855
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 6843 . . . . . 6 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
6 cbvixp.3 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
75, 6eleq12d 2828 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝑓𝑦) ∈ 𝐶))
82, 4, 7cbvralw 3288 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)
98anbi2i 624 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶))
109abbii 2803 . 2 {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
11 dfixp 8840 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
12 dfixp 8840 . 2 X𝑦𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
1310, 11, 123eqtr4i 2771 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wnfc 2884  wral 3061   Fn wfn 6492  cfv 6497  Xcixp 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fn 6500  df-fv 6505  df-ixp 8839
This theorem is referenced by:  cbvixpv  8856  mptelixpg  8876  ixpiunwdom  9531  prdsbas3  17368  elptr2  22941  ptunimpt  22962  ptcldmpt  22981  finixpnum  36109  ptrest  36123  hoimbl2  44992  vonhoire  44999  vonn0ioo2  45017  vonn0icc2  45019
  Copyright terms: Public domain W3C validator