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Mirrors > Home > MPE Home > Th. List > xpmapen | Structured version Visualization version GIF version |
Description: Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
xpmapen.1 | ⊢ 𝐴 ∈ V |
xpmapen.2 | ⊢ 𝐵 ∈ V |
xpmapen.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
xpmapen | ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpmapen.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | xpmapen.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | xpmapen.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | 2fveq3 6438 | . . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧))) | |
5 | 4 | cbvmptv 4973 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) |
6 | 2fveq3 6438 | . . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧))) | |
7 | 6 | cbvmptv 4973 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) |
8 | fveq2 6433 | . . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧)) | |
9 | fveq2 6433 | . . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧)) | |
10 | 8, 9 | opeq12d 4631 | . . 3 ⊢ (𝑤 = 𝑧 → 〈((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)〉 = 〈((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)〉) |
11 | 10 | cbvmptv 4973 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ 〈((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)〉) = (𝑧 ∈ 𝐶 ↦ 〈((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)〉) |
12 | 1, 2, 3, 5, 7, 11 | xpmapenlem 8396 | 1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 Vcvv 3414 〈cop 4403 class class class wbr 4873 ↦ cmpt 4952 × cxp 5340 ‘cfv 6123 (class class class)co 6905 1st c1st 7426 2nd c2nd 7427 ↑𝑚 cmap 8122 ≈ cen 8219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-map 8124 df-en 8223 |
This theorem is referenced by: rexpen 15331 |
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