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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 6994 | . 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ 𝑋) ∈ V) | |
2 | f1oi 6086 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6057 | . . 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 8359 | . 2 ⊢ ((( I ↾ 𝑋) ∈ V ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → 𝑋 ≼* 𝑋) | |
6 | 1, 4, 5 | sylancl 693 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 I cid 4948 ↾ cres 5040 –onto→wfo 5802 –1-1-onto→wf1o 5803 ≼* 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-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-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-wdom 8347 |
This theorem is referenced by: hsmexlem3 9133 hsmexlem5 9135 |
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