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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 7692 | . 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ 𝑋) ∈ V) | |
2 | f1oi 6698 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6668 | . . 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 9187 | . 2 ⊢ ((( I ↾ 𝑋) ∈ V ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → 𝑋 ≼* 𝑋) | |
6 | 1, 4, 5 | sylancl 589 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3408 class class class wbr 5053 I cid 5454 ↾ cres 5553 –onto→wfo 6378 –1-1-onto→wf1o 6379 ≼* cwdom 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-wdom 9181 |
This theorem is referenced by: hsmexlem3 10042 hsmexlem5 10044 |
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