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Mirrors > Home > MPE Home > Th. List > Mathboxes > ixpeq1i | Structured version Visualization version GIF version |
Description: Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
ixpeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
ixpeq1i | ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpeq1i.1 | . . . . . . 7 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2829 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | abbii 2805 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝑥 ∈ 𝐵} |
4 | 3 | fneq2i 6663 | . . . 4 ⊢ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵}) |
5 | 2 | imbi1i 349 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶)) |
6 | 5 | ralbii2 3085 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶) |
7 | 4, 6 | anbi12i 627 | . . 3 ⊢ ((𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)) |
8 | 7 | abbii 2805 | . 2 ⊢ {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)} |
9 | df-ixp 8932 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} | |
10 | df-ixp 8932 | . 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)} | |
11 | 8, 9, 10 | 3eqtr4i 2771 | 1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1535 ∈ wcel 2104 {cab 2710 ∀wral 3057 Fn wfn 6554 ‘cfv 6559 Xcixp 8931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-fn 6562 df-ixp 8932 |
This theorem is referenced by: ixpeq12i 36143 |
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