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

Theorem cbvixp 8078
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 2930 . . . . 5 𝑦(𝑓𝑥) ∈ 𝐵
3 cbvixp.2 . . . . . 6 𝑥𝐶
43nfel2 2930 . . . . 5 𝑥(𝑓𝑦) ∈ 𝐶
5 fveq2 6332 . . . . . 6 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
6 cbvixp.3 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
75, 6eleq12d 2844 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝑓𝑦) ∈ 𝐶))
82, 4, 7cbvral 3316 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)
98anbi2i 601 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶))
109abbii 2888 . 2 {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
11 dfixp 8063 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
12 dfixp 8063 . 2 X𝑦𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
1310, 11, 123eqtr4i 2803 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wnfc 2900  wral 3061   Fn wfn 6026  cfv 6031  Xcixp 8061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fn 6034  df-fv 6039  df-ixp 8062
This theorem is referenced by:  cbvixpv  8079  mptelixpg  8098  ixpiunwdom  8651  prdsbas3  16348  elptr2  21597  ptunimpt  21618  ptcldmpt  21637  finixpnum  33723  ptrest  33737  hoimbl2  41395  vonhoire  41402  vonn0ioo2  41420  vonn0icc2  41422
  Copyright terms: Public domain W3C validator