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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 2832 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
| 3 | 2 | abbii 2808 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝑥 ∈ 𝐵} |
| 4 | 3 | fneq2i 6664 | . . . 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 3088 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶) |
| 7 | 4, 6 | anbi12i 628 | . . 3 ⊢ ((𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)) |
| 8 | 7 | abbii 2808 | . 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 8934 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} | |
| 10 | df-ixp 8934 | . 2 ⊢ X𝑥 ∈ 𝐵 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐵} ∧ ∀𝑥 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)} | |
| 11 | 8, 9, 10 | 3eqtr4i 2774 | 1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2713 ∀wral 3060 Fn wfn 6554 ‘cfv 6559 Xcixp 8933 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-fn 6562 df-ixp 8934 |
| This theorem is referenced by: ixpeq12i 36180 |
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