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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 2919 | . . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵 |
3 | cbvixp.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfel2 2919 | . . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶 |
5 | fveq2 6892 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦)) | |
6 | cbvixp.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
7 | 5, 6 | eleq12d 2825 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶)) |
8 | 2, 4, 7 | cbvralw 3301 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶) |
9 | 8 | anbi2i 621 | . . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)) |
10 | 9 | abbii 2800 | . 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} |
11 | dfixp 8897 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
12 | dfixp 8897 | . 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} | |
13 | 10, 11, 12 | 3eqtr4i 2768 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {cab 2707 Ⅎwnfc 2881 ∀wral 3059 Fn wfn 6539 ‘cfv 6544 Xcixp 8895 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fn 6547 df-fv 6552 df-ixp 8896 |
This theorem is referenced by: cbvixpv 8913 mptelixpg 8933 ixpiunwdom 9589 prdsbas3 17433 elptr2 23300 ptunimpt 23321 ptcldmpt 23340 finixpnum 36778 ptrest 36792 hoimbl2 45681 vonhoire 45688 vonn0ioo2 45706 vonn0icc2 45708 |
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