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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 8925 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | domwdom 9518 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵) |
5 | wdomtr 9519 | . . 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 8949 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
9 | endom 8925 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
10 | domwdom 9518 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴) |
12 | wdomtr 9519 | . . 3 ⊢ ((𝐶 ≼* 𝐵 ∧ 𝐵 ≼* 𝐴) → 𝐶 ≼* 𝐴) | |
13 | 7, 11, 12 | syl2anr 598 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐵) → 𝐶 ≼* 𝐴) |
14 | 6, 13 | impbida 800 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 class class class wbr 5109 ≈ cen 8886 ≼ cdom 8887 ≼* cwdom 9508 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-wdom 9509 |
This theorem is referenced by: (None) |
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