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| Mirrors > Home > MPE Home > Th. List > cbvixp | Structured version Visualization version GIF version | ||
| Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.) |
| Ref | Expression |
|---|---|
| cbvixp.1 | ⊢ Ⅎ𝑦𝐵 |
| cbvixp.2 | ⊢ Ⅎ𝑥𝐶 |
| cbvixp.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| cbvixp | ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvixp.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
| 2 | 1 | nfel2 2924 | . . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵 |
| 3 | cbvixp.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
| 4 | 3 | nfel2 2924 | . . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶 |
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦)) | |
| 6 | cbvixp.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 7 | 5, 6 | eleq12d 2835 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶)) |
| 8 | 2, 4, 7 | cbvralw 3306 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶) |
| 9 | 8 | anbi2i 623 | . . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)) |
| 10 | 9 | abbii 2809 | . 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} |
| 11 | dfixp 8939 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
| 12 | dfixp 8939 | . 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} | |
| 13 | 10, 11, 12 | 3eqtr4i 2775 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∀wral 3061 Fn wfn 6556 ‘cfv 6561 Xcixp 8937 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fn 6564 df-fv 6569 df-ixp 8938 |
| This theorem is referenced by: mptelixpg 8975 ixpiunwdom 9630 prdsbas3 17526 elptr2 23582 ptunimpt 23603 ptcldmpt 23622 finixpnum 37612 ptrest 37626 hoimbl2 46680 vonhoire 46687 vonn0ioo2 46705 vonn0icc2 46707 |
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