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Mirrors > Home > MPE Home > Th. List > wdomen2 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
wdomen2 | ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐶 ≼* 𝐴 → 𝐶 ≼* 𝐴) | |
2 | endom 8530 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | domwdom 9032 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵) |
5 | wdomtr 9033 | . . 3 ⊢ ((𝐶 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → 𝐶 ≼* 𝐵) | |
6 | 1, 4, 5 | syl2anr 598 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐴) → 𝐶 ≼* 𝐵) |
7 | id 22 | . . 3 ⊢ (𝐶 ≼* 𝐵 → 𝐶 ≼* 𝐵) | |
8 | ensym 8552 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
9 | endom 8530 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
10 | domwdom 9032 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴) |
12 | wdomtr 9033 | . . 3 ⊢ ((𝐶 ≼* 𝐵 ∧ 𝐵 ≼* 𝐴) → 𝐶 ≼* 𝐴) | |
13 | 7, 11, 12 | syl2anr 598 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐵) → 𝐶 ≼* 𝐴) |
14 | 6, 13 | impbida 799 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 class class class wbr 5058 ≈ cen 8500 ≼ cdom 8501 ≼* cwdom 9015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-wdom 9017 |
This theorem is referenced by: (None) |
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