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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 6430 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑓 Fn 𝐴 ↔ 𝑓 Fn 𝐵)) | |
2 | raleq 3310 | . . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)) | |
3 | 1, 2 | anbi12d 634 | . . 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 8509 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} | |
6 | dfixp 8509 | . 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 399 = wceq 1542 ∈ wcel 2114 {cab 2716 ∀wral 3053 Fn wfn 6334 ‘cfv 6339 Xcixp 8507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-fn 6342 df-ixp 8508 |
This theorem is referenced by: ixpeq1d 8519 finixpnum 35385 ioorrnopn 43388 ioorrnopnxr 43390 ovnval 43621 hoicvr 43628 hoidmv1le 43674 hoidmvle 43680 ovnhoi 43683 hspval 43689 ovnlecvr2 43690 hoiqssbl 43705 vonhoire 43752 iunhoiioo 43756 vonioo 43762 vonicc 43765 |
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