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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 7621 | . 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ 𝑋) ∈ V) | |
2 | f1oi 6654 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6624 | . . 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 9037 | . 2 ⊢ ((( I ↾ 𝑋) ∈ V ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → 𝑋 ≼* 𝑋) | |
6 | 1, 4, 5 | sylancl 588 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 I cid 5461 ↾ cres 5559 –onto→wfo 6355 –1-1-onto→wf1o 6356 ≼* cwdom 9023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-wdom 9025 |
This theorem is referenced by: hsmexlem3 9852 hsmexlem5 9854 |
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