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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 9606 | . . . . . . 7 ⊢ Rel ≼* | |
| 2 | 1 | brrelex2i 5742 | . . . . . 6 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
| 3 | 0domg 9140 | . . . . . 6 ⊢ (𝑌 ∈ V → ∅ ≼ 𝑌) | |
| 4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → ∅ ≼ 𝑌) |
| 5 | breq1 5146 | . . . . 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 9609 | . . . . 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 3484 | . . . . . . . . . 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 10100 | . . . . . . . 8 ⊢ ((𝑌 ∈ V ∧ 𝑋 ∈ Fin ∧ 𝑥:𝑌–onto→𝑋) → 𝑋 ≼ 𝑌) | |
| 18 | 14, 15, 16, 17 | syl3anc 1373 | . . . . . . 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 1933 | . . . 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 3029 | . 2 ⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 → 𝑋 ≼ 𝑌)) |
| 24 | domwdom 9614 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ⟶wf 6557 –onto→wfo 6559 ≼ cdom 8983 Fincfn 8985 ≼* cwdom 9604 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-wdom 9605 df-card 9979 df-acn 9982 |
| This theorem is referenced by: (None) |
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