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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 2996 | . . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵 |
3 | cbvixp.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfel2 2996 | . . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶 |
5 | fveq2 6664 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦)) | |
6 | cbvixp.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
7 | 5, 6 | eleq12d 2907 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶)) |
8 | 2, 4, 7 | cbvralw 3441 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶) |
9 | 8 | anbi2i 624 | . . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)) |
10 | 9 | abbii 2886 | . 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} |
11 | dfixp 8457 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
12 | dfixp 8457 | . 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} | |
13 | 10, 11, 12 | 3eqtr4i 2854 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 Fn wfn 6344 ‘cfv 6349 Xcixp 8455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fn 6352 df-fv 6357 df-ixp 8456 |
This theorem is referenced by: cbvixpv 8473 mptelixpg 8493 ixpiunwdom 9049 prdsbas3 16748 elptr2 22176 ptunimpt 22197 ptcldmpt 22216 finixpnum 34871 ptrest 34885 hoimbl2 42941 vonhoire 42948 vonn0ioo2 42966 vonn0icc2 42968 |
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