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Mirrors > Home > MPE Home > Th. List > xpen | Structured version Visualization version GIF version |
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
xpen | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8113 | . . . . 5 ⊢ Rel ≈ | |
2 | 1 | brrelexi 5298 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V) |
3 | endom 8135 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
4 | xpdom1g 8212 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
5 | 2, 3, 4 | syl2anr 576 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
6 | 1 | brrelex2i 5299 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | endom 8135 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≼ 𝐷) | |
8 | xpdom2g 8211 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ≼ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) | |
9 | 6, 7, 8 | syl2an 575 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) |
10 | domtr 8161 | . . 3 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) | |
11 | 5, 9, 10 | syl2anc 565 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) |
12 | 1 | brrelex2i 5299 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
13 | ensym 8157 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
14 | endom 8135 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
16 | xpdom1g 8212 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐵 ≼ 𝐴) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) | |
17 | 12, 15, 16 | syl2anr 576 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) |
18 | 1 | brrelexi 5298 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
19 | ensym 8157 | . . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ 𝐶) | |
20 | endom 8135 | . . . . 5 ⊢ (𝐷 ≈ 𝐶 → 𝐷 ≼ 𝐶) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≼ 𝐶) |
22 | xpdom2g 8211 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐷 ≼ 𝐶) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) | |
23 | 18, 21, 22 | syl2an 575 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) |
24 | domtr 8161 | . . 3 ⊢ (((𝐵 × 𝐷) ≼ (𝐴 × 𝐷) ∧ (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) | |
25 | 17, 23, 24 | syl2anc 565 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) |
26 | sbth 8235 | . 2 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐷) ∧ (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | |
27 | 11, 25, 26 | syl2anc 565 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2144 Vcvv 3349 class class class wbr 4784 × cxp 5247 ≈ cen 8105 ≼ cdom 8106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-1st 7314 df-2nd 7315 df-er 7895 df-en 8109 df-dom 8110 |
This theorem is referenced by: map2xp 8285 unxpdom2 8323 sucxpdom 8324 xpnum 8976 infxpenlem 9035 infxpidm2 9039 xpcdaen 9206 mapcdaen 9207 pwcdaen 9208 cdaxpdom 9212 ackbij1lem5 9247 canthp1lem1 9675 xpnnen 15144 qnnen 15147 rexpen 15162 met2ndci 22546 re2ndc 22823 dyadmbl 23587 opnmblALT 23590 mbfimaopnlem 23641 mblfinlem1 33772 |
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