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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 6848 | . . 3 ⊢ (𝑤 = 𝑧 → (1st ‘(𝑥‘𝑤)) = (1st ‘(𝑥‘𝑧))) | |
5 | 4 | cbvmptv 5219 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) |
6 | 2fveq3 6848 | . . 3 ⊢ (𝑤 = 𝑧 → (2nd ‘(𝑥‘𝑤)) = (2nd ‘(𝑥‘𝑧))) | |
7 | 6 | cbvmptv 5219 | . 2 ⊢ (𝑤 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑤))) = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) |
8 | fveq2 6843 | . . . 4 ⊢ (𝑤 = 𝑧 → ((1st ‘𝑦)‘𝑤) = ((1st ‘𝑦)‘𝑧)) | |
9 | fveq2 6843 | . . . 4 ⊢ (𝑤 = 𝑧 → ((2nd ‘𝑦)‘𝑤) = ((2nd ‘𝑦)‘𝑧)) | |
10 | 8, 9 | opeq12d 4839 | . . 3 ⊢ (𝑤 = 𝑧 → ⟨((1st ‘𝑦)‘𝑤), ((2nd ‘𝑦)‘𝑤)⟩ = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) |
11 | 10 | cbvmptv 5219 | . 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 9091 | 1 ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3444 ⟨cop 4593 class class class wbr 5106 ↦ cmpt 5189 × cxp 5632 ‘cfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 ↑m cmap 8768 ≈ cen 8883 |
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 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-en 8887 |
This theorem is referenced by: rexpen 16115 |
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