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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvixpvw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in an indexed Cartesian product, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvixpvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
cbvixpvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvixpvw2 | ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | cbvixpvw2.2 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
3 | 1, 2 | eleq12d 2834 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 3 | cbvabv 2811 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐵} |
5 | 4 | fneq2i 6664 | . . . 4 ⊢ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵}) |
6 | fveq2 6904 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡‘𝑥) = (𝑡‘𝑦)) | |
7 | cbvixpvw2.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
8 | 6, 7 | eleq12d 2834 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑡‘𝑥) ∈ 𝐶 ↔ (𝑡‘𝑦) ∈ 𝐷)) |
9 | 2, 8 | cbvralvw2 36205 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷) |
10 | 5, 9 | anbi12i 628 | . . 3 ⊢ ((𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)) |
11 | 10 | abbii 2808 | . 2 ⊢ {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)} |
12 | df-ixp 8934 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)} | |
13 | df-ixp 8934 | . 2 ⊢ X𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)} | |
14 | 11, 12, 13 | 3eqtr4i 2774 | 1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fn 6562 df-fv 6567 df-ixp 8934 |
This theorem is referenced by: (None) |
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