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Mirrors > Home > MPE Home > Th. List > wdomref | Structured version Visualization version GIF version |
Description: Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
wdomref | ⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 7935 | . 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ 𝑋) ∈ V) | |
2 | f1oi 6887 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6856 | . . 3 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–onto→𝑋) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ( I ↾ 𝑋):𝑋–onto→𝑋 |
5 | fowdom 9609 | . 2 ⊢ ((( I ↾ 𝑋) ∈ V ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → 𝑋 ≼* 𝑋) | |
6 | 1, 4, 5 | sylancl 586 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 I cid 5582 ↾ cres 5691 –onto→wfo 6561 –1-1-onto→wf1o 6562 ≼* cwdom 9602 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-wdom 9603 |
This theorem is referenced by: hsmexlem3 10466 hsmexlem5 10468 |
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