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Mirrors > Home > MPE Home > Th. List > wdomfil | Structured version Visualization version GIF version |
Description: Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
wdomfil | ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 9014 | . . . . . . 7 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5573 | . . . . . 6 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 0domg 8628 | . . . . . 6 ⊢ (𝑌 ∈ V → ∅ ≼ 𝑌) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → ∅ ≼ 𝑌) |
5 | breq1 5033 | . . . . 5 ⊢ (𝑋 = ∅ → (𝑋 ≼ 𝑌 ↔ ∅ ≼ 𝑌)) | |
6 | 4, 5 | syl5ibr 249 | . . . 4 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
7 | 6 | adantl 485 | . . 3 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 = ∅) → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
8 | brwdomn0 9017 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋)) | |
9 | 8 | adantl 485 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋)) |
10 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
11 | fof 6565 | . . . . . . . . . 10 ⊢ (𝑥:𝑌–onto→𝑋 → 𝑥:𝑌⟶𝑋) | |
12 | dmfex 7623 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑥:𝑌⟶𝑋) → 𝑌 ∈ V) | |
13 | 10, 11, 12 | sylancr 590 | . . . . . . . . 9 ⊢ (𝑥:𝑌–onto→𝑋 → 𝑌 ∈ V) |
14 | 13 | adantl 485 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑌 ∈ V) |
15 | simpl 486 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ∈ Fin) | |
16 | simpr 488 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑥:𝑌–onto→𝑋) | |
17 | fodomfi2 9471 | . . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ 𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ≼ 𝑌) | |
18 | 14, 15, 16, 17 | syl3anc 1368 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ≼ 𝑌) |
19 | 18 | ex 416 | . . . . . 6 ⊢ (𝑋 ∈ Fin → (𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
20 | 19 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
21 | 20 | exlimdv 1934 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
22 | 9, 21 | sylbid 243 | . . 3 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
23 | 7, 22 | pm2.61dane 3074 | . 2 ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
24 | domwdom 9022 | . 2 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) | |
25 | 23, 24 | impbid1 228 | 1 ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ⟶wf 6320 –onto→wfo 6322 ≼ cdom 8490 Fincfn 8492 ≼* cwdom 9012 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-1o 8085 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wdom 9013 df-card 9352 df-acn 9355 |
This theorem is referenced by: (None) |
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