![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ixpeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
Ref | Expression |
---|---|
ixpeq1 | ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq2 6595 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Fn 𝐴 ↔ 𝑓 Fn 𝐵)) | |
2 | raleq 3308 | . . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)) | |
3 | 1, 2 | anbi12d 632 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))) |
4 | 3 | abbidv 2802 | . 2 ⊢ (𝐴 = 𝐵 → {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)}) |
5 | dfixp 8840 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} | |
6 | dfixp 8840 | . 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)} | |
7 | 4, 5, 6 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-fn 6500 df-ixp 8839 |
This theorem is referenced by: ixpeq1d 8850 finixpnum 36109 ioorrnopn 44632 ioorrnopnxr 44634 ovnval 44868 hoicvr 44875 hoidmv1le 44921 hoidmvle 44927 ovnhoi 44930 hspval 44936 ovnlecvr2 44937 hoiqssbl 44952 vonhoire 44999 iunhoiioo 45003 vonioo 45009 vonicc 45012 |
Copyright terms: Public domain | W3C validator |