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Theorem domdifsn 8084
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 8025 . . . . 5 (𝐴𝐵𝐴𝐵)
2 relsdom 8004 . . . . . . 7 Rel ≺
32brrelex2i 5193 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4 brdomg 8007 . . . . . 6 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
53, 4syl 17 . . . . 5 (𝐴𝐵 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
61, 5mpbid 222 . . . 4 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
76adantr 480 . . 3 ((𝐴𝐵𝐶𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
8 f1f 6139 . . . . . . . 8 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
9 frn 6091 . . . . . . . 8 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
108, 9syl 17 . . . . . . 7 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1110adantl 481 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
12 sdomnen 8026 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1312ad2antrr 762 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
14 vex 3234 . . . . . . . . . . 11 𝑓 ∈ V
15 dff1o5 6184 . . . . . . . . . . . 12 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
1615biimpri 218 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵) → 𝑓:𝐴1-1-onto𝐵)
17 f1oen3g 8013 . . . . . . . . . . 11 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
1814, 16, 17sylancr 696 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴𝐵)
1918ex 449 . . . . . . . . 9 (𝑓:𝐴1-1𝐵 → (ran 𝑓 = 𝐵𝐴𝐵))
2019necon3bd 2837 . . . . . . . 8 (𝑓:𝐴1-1𝐵 → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2120adantl 481 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2213, 21mpd 15 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
23 pssdifn0 3977 . . . . . 6 ((ran 𝑓𝐵 ∧ ran 𝑓𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
2411, 22, 23syl2anc 694 . . . . 5 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
25 n0 3964 . . . . 5 ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
2624, 25sylib 208 . . . 4 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
272brrelexi 5192 . . . . . . . . 9 (𝐴𝐵𝐴 ∈ V)
2827ad2antrr 762 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
293ad2antrr 762 . . . . . . . . 9 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
30 difexg 4841 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ {𝑥}) ∈ V)
3129, 30syl 17 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
32 eldifn 3766 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
33 disjsn 4278 . . . . . . . . . . . . 13 ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓)
3432, 33sylibr 224 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅)
3534adantl 481 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅)
3610adantr 480 . . . . . . . . . . . 12 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓𝐵)
37 reldisj 4053 . . . . . . . . . . . 12 (ran 𝑓𝐵 → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
3836, 37syl 17 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})))
3935, 38mpbid 222 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓 ⊆ (𝐵 ∖ {𝑥}))
40 f1ssr 6145 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4139, 40syldan 486 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4241adantl 481 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
43 f1dom2g 8015 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
4428, 31, 42, 43syl3anc 1366 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
45 eldifi 3765 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥𝐵)
4645ad2antll 765 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥𝐵)
47 simplr 807 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶𝐵)
48 difsnen 8083 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑥𝐵𝐶𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
4929, 46, 47, 48syl3anc 1366 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
50 domentr 8056 . . . . . . 7 ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
5144, 49, 50syl2anc 694 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
5251expr 642 . . . . 5 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
5352exlimdv 1901 . . . 4 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝐴 ≼ (𝐵 ∖ {𝐶})))
5426, 53mpd 15 . . 3 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
557, 54exlimddv 1903 . 2 ((𝐴𝐵𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
561adantr 480 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴𝐵)
57 difsn 4360 . . . . 5 𝐶𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
5857breq2d 4697 . . . 4 𝐶𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
5958adantl 481 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
6056, 59mpbird 247 . 2 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
6155, 60pm2.61dan 849 1 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  ran crn 5144  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cen 7994  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  domunsn  8151  marypha1lem  8380
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