Theorem ixpeq1 6533

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq1	⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ixpeq1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fneq2 5148	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Fn 𝐴 ↔ 𝑓 Fn 𝐵))
2		raleq 2584	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
3	1, 2	anbi12d 460	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)))
4	3	abbidv 2217	. 2 ⊢ (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)})
5		dfixp 6524	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
6		dfixp 6524	. 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}
7	4, 5, 6	3eqtr4g 2157	1 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448  {cab 2086  ∀wral 2375   Fn wfn 5054  ‘cfv 5059  Xcixp 6522

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-fn 5062  df-ixp 6523

This theorem is referenced by:  ixpeq1d  6534