MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domdifsn Structured version   Visualization version   GIF version

Theorem domdifsn 8588
Description: Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domdifsn (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))

Proof of Theorem domdifsn
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 8525 . . . . 5 (𝐴𝐵𝐴𝐵)
2 relsdom 8504 . . . . . . 7 Rel ≺
32brrelex2i 5602 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4 brdomg 8507 . . . . . 6 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
53, 4syl 17 . . . . 5 (𝐴𝐵 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
61, 5mpbid 233 . . . 4 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
76adantr 481 . . 3 ((𝐴𝐵𝐶𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
8 f1f 6568 . . . . . . . 8 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
98frnd 6514 . . . . . . 7 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
109adantl 482 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
11 sdomnen 8526 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1211ad2antrr 722 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
13 vex 3495 . . . . . . . . . . 11 𝑓 ∈ V
14 dff1o5 6617 . . . . . . . . . . . 12 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
1514biimpri 229 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴1-1-onto𝐵)
16 f1oen3g 8513 . . . . . . . . . . 11 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
1713, 15, 16sylancr 587 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴𝐵)
1817ex 413 . . . . . . . . 9 (𝑓:𝐴1-1𝐵 → (ran 𝑓 = 𝐵𝐴𝐵))
1918necon3bd 3027 . . . . . . . 8 (𝑓:𝐴1-1𝐵 → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2019adantl 482 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2112, 20mpd 15 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
22 pssdifn0 4322 . . . . . 6 ((ran 𝑓𝐵 ∧ ran 𝑓𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
2310, 21, 22syl2anc 584 . . . . 5 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
24 n0 4307 . . . . 5 ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
2523, 24sylib 219 . . . 4 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
262brrelex1i 5601 . . . . . . . . 9 (𝐴𝐵𝐴 ∈ V)
2726ad2antrr 722 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
283ad2antrr 722 . . . . . . . . 9 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
29 difexg 5222 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ {𝑥}) ∈ V)
3028, 29syl 17 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
31 eldifn 4101 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
32 disjsn 4639 . . . . . . . . . . . . 13 ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓)
3331, 32sylibr 235 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅)
3433adantl 482 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅)
359adantr 481 . . . . . . . . . . . 12 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓𝐵)
36 reldisj 4398 . . . . . . . . . . . 12 (ran 𝑓𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
3735, 36syl 17 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
3834, 37mpbid 233 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))
39 f1ssr 6574 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4038, 39syldan 591 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4140adantl 482 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
42 f1dom2g 8515 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
4327, 30, 41, 42syl3anc 1363 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
44 eldifi 4100 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥𝐵)
4544ad2antll 725 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥𝐵)
46 simplr 765 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶𝐵)
47 difsnen 8587 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑥𝐵𝐶𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
4828, 45, 46, 47syl3anc 1363 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
49 domentr 8556 . . . . . . 7 ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
5043, 48, 49syl2anc 584 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
5150expr 457 . . . . 5 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
5251exlimdv 1925 . . . 4 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
5325, 52mpd 15 . . 3 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
547, 53exlimddv 1927 . 2 ((𝐴𝐵𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
551adantr 481 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴𝐵)
56 difsn 4723 . . . . 5 𝐶𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
5756breq2d 5069 . . . 4 𝐶𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
5857adantl 482 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
5955, 58mpbird 258 . 2 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
6054, 59pm2.61dan 809 1 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  {csn 4557   class class class wbr 5057  ran crn 5549  1-1wf1 6345  1-1-ontowf1o 6347  cen 8494  cdom 8495  csdm 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-suc 6190  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500
This theorem is referenced by:  domunsn  8655  marypha1lem  8885
  Copyright terms: Public domain W3C validator