Theorem ixpeq1 8902

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 6627	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Fn 𝐴 ↔ 𝑓 Fn 𝐵))
2		raleq 3320	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
3	1, 2	anbi12d 643	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)))
4	3	abbidv 2829	. 2 ⊢ (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)})
5		dfixp 8893	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
6		dfixp 8893	. 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}
7	4, 5, 6	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079   Fn wfn 6531  ‘cfv 6536  Xcixp 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-fn 6539  df-ixp 8892

This theorem is referenced by:  ixpeq1d  8903  finixpnum  38256  ioorrnopn  47019  ioorrnopnxr  47021  ovnval  47255  hoicvr  47262  hoidmv1le  47308  hoidmvle  47314  ovnhoi  47317  hspval  47323  ovnlecvr2  47324  hoiqssbl  47339  vonhoire  47386  iunhoiioo  47390  vonioo  47396  vonicc  47399