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Theorem cbvixp 8954
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 2924 . . . . 5 𝑦(𝑓𝑥) ∈ 𝐵
3 cbvixp.2 . . . . . 6 𝑥𝐶
43nfel2 2924 . . . . 5 𝑥(𝑓𝑦) ∈ 𝐶
5 fveq2 6906 . . . . . 6 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
6 cbvixp.3 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
75, 6eleq12d 2835 . . . . 5 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝐵 ↔ (𝑓𝑦) ∈ 𝐶))
82, 4, 7cbvralw 3306 . . . 4 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)
98anbi2i 623 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶))
109abbii 2809 . 2 {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
11 dfixp 8939 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
12 dfixp 8939 . 2 X𝑦𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝐶)}
1310, 11, 123eqtr4i 2775 1 X𝑥𝐴 𝐵 = X𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wnfc 2890  wral 3061   Fn wfn 6556  cfv 6561  Xcixp 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fn 6564  df-fv 6569  df-ixp 8938
This theorem is referenced by:  mptelixpg  8975  ixpiunwdom  9630  prdsbas3  17526  elptr2  23582  ptunimpt  23603  ptcldmpt  23622  finixpnum  37612  ptrest  37626  hoimbl2  46680  vonhoire  46687  vonn0ioo2  46705  vonn0icc2  46707
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