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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 7868 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | domwdom 8362 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵) |
5 | wdomtr 8363 | . . 3 ⊢ ((𝐶 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → 𝐶 ≼* 𝐵) | |
6 | 1, 4, 5 | syl2anr 494 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐴) → 𝐶 ≼* 𝐵) |
7 | id 22 | . . 3 ⊢ (𝐶 ≼* 𝐵 → 𝐶 ≼* 𝐵) | |
8 | ensym 7891 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
9 | endom 7868 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
10 | domwdom 8362 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴) |
12 | wdomtr 8363 | . . 3 ⊢ ((𝐶 ≼* 𝐵 ∧ 𝐵 ≼* 𝐴) → 𝐶 ≼* 𝐴) | |
13 | 7, 11, 12 | syl2anr 494 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐵) → 𝐶 ≼* 𝐴) |
14 | 6, 13 | impbida 873 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 ≼* cwdom 8345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-wdom 8347 |
This theorem is referenced by: (None) |
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