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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 7847 | . 2 ⊢ (𝑋 ∈ 𝑉 → ( I ↾ 𝑋) ∈ V) | |
2 | f1oi 6819 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6788 | . . 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 9503 | . 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 3443 class class class wbr 5103 I cid 5528 ↾ cres 5633 –onto→wfo 6491 –1-1-onto→wf1o 6492 ≼* cwdom 9496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-wdom 9497 |
This theorem is referenced by: hsmexlem3 10360 hsmexlem5 10362 |
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