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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 2922 | . . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵 |
3 | cbvixp.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfel2 2922 | . . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶 |
5 | fveq2 6843 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦)) | |
6 | cbvixp.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
7 | 5, 6 | eleq12d 2828 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶)) |
8 | 2, 4, 7 | cbvralw 3288 | . . . 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 2803 | . 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} |
11 | dfixp 8840 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
12 | dfixp 8840 | . 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)} | |
13 | 10, 11, 12 | 3eqtr4i 2771 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3061 Fn wfn 6492 ‘cfv 6497 Xcixp 8838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fn 6500 df-fv 6505 df-ixp 8839 |
This theorem is referenced by: cbvixpv 8856 mptelixpg 8876 ixpiunwdom 9531 prdsbas3 17368 elptr2 22941 ptunimpt 22962 ptcldmpt 22981 finixpnum 36109 ptrest 36123 hoimbl2 44992 vonhoire 44999 vonn0ioo2 45017 vonn0icc2 45019 |
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