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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 9603 | . . . . . . 7 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5745 | . . . . . 6 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 0domg 9138 | . . . . . 6 ⊢ (𝑌 ∈ V → ∅ ≼ 𝑌) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → ∅ ≼ 𝑌) |
5 | breq1 5150 | . . . . 5 ⊢ (𝑋 = ∅ → (𝑋 ≼ 𝑌 ↔ ∅ ≼ 𝑌)) | |
6 | 4, 5 | imbitrrid 246 | . . . 4 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
7 | 6 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 = ∅) → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
8 | brwdomn0 9606 | . . . . 5 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋)) | |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 ↔ ∃𝑥 𝑥:𝑌–onto→𝑋)) |
10 | vex 3481 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
11 | fof 6820 | . . . . . . . . . 10 ⊢ (𝑥:𝑌–onto→𝑋 → 𝑥:𝑌⟶𝑋) | |
12 | dmfex 7927 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ V ∧ 𝑥:𝑌⟶𝑋) → 𝑌 ∈ V) | |
13 | 10, 11, 12 | sylancr 587 | . . . . . . . . 9 ⊢ (𝑥:𝑌–onto→𝑋 → 𝑌 ∈ V) |
14 | 13 | adantl 481 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑌 ∈ V) |
15 | simpl 482 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ∈ Fin) | |
16 | simpr 484 | . . . . . . . 8 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑥:𝑌–onto→𝑋) | |
17 | fodomfi2 10097 | . . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ 𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ≼ 𝑌) | |
18 | 14, 15, 16, 17 | syl3anc 1370 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ≼ 𝑌) |
19 | 18 | ex 412 | . . . . . 6 ⊢ (𝑋 ∈ Fin → (𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
21 | 20 | exlimdv 1930 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (∃𝑥 𝑥:𝑌–onto→𝑋 → 𝑋 ≼ 𝑌)) |
22 | 9, 21 | sylbid 240 | . . 3 ⊢ ((𝑋 ∈ Fin ∧ 𝑋 ≠ ∅) → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
23 | 7, 22 | pm2.61dane 3026 | . 2 ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
24 | domwdom 9611 | . 2 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) | |
25 | 23, 24 | impbid1 225 | 1 ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 –onto→wfo 6560 ≼ cdom 8981 Fincfn 8983 ≼* cwdom 9601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-wdom 9602 df-card 9976 df-acn 9979 |
This theorem is referenced by: (None) |
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