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Theorem domdifsn 7988
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 7928 . . . . 5 (𝐴𝐵𝐴𝐵)
2 relsdom 7907 . . . . . . 7 Rel ≺
32brrelex2i 5124 . . . . . 6 (𝐴𝐵𝐵 ∈ V)
4 brdomg 7910 . . . . . 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 481 . . 3 ((𝐴𝐵𝐶𝐵) → ∃𝑓 𝑓:𝐴1-1𝐵)
8 f1f 6060 . . . . . . . 8 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
9 frn 6012 . . . . . . . 8 (𝑓:𝐴𝐵 → ran 𝑓𝐵)
108, 9syl 17 . . . . . . 7 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1110adantl 482 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
12 sdomnen 7929 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1312ad2antrr 761 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
14 vex 3194 . . . . . . . . . . 11 𝑓 ∈ V
15 dff1o5 6105 . . . . . . . . . . . 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 7916 . . . . . . . . . . 11 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
1814, 16, 17sylancr 694 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵) → 𝐴𝐵)
1918ex 450 . . . . . . . . 9 (𝑓:𝐴1-1𝐵 → (ran 𝑓 = 𝐵𝐴𝐵))
2019necon3bd 2810 . . . . . . . 8 (𝑓:𝐴1-1𝐵 → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2120adantl 482 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (¬ 𝐴𝐵 → ran 𝑓𝐵))
2213, 21mpd 15 . . . . . 6 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
23 pssdifn0 3923 . . . . . 6 ((ran 𝑓𝐵 ∧ ran 𝑓𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
2411, 22, 23syl2anc 692 . . . . 5 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → (𝐵 ∖ ran 𝑓) ≠ ∅)
25 n0 3912 . . . . 5 ((𝐵 ∖ ran 𝑓) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
2624, 25sylib 208 . . . 4 (((𝐴𝐵𝐶𝐵) ∧ 𝑓:𝐴1-1𝐵) → ∃𝑥 𝑥 ∈ (𝐵 ∖ ran 𝑓))
272brrelexi 5123 . . . . . . . . 9 (𝐴𝐵𝐴 ∈ V)
2827ad2antrr 761 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ∈ V)
293ad2antrr 761 . . . . . . . . 9 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐵 ∈ V)
30 difexg 4773 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ {𝑥}) ∈ V)
3129, 30syl 17 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ∈ V)
32 eldifn 3716 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → ¬ 𝑥 ∈ ran 𝑓)
33 disjsn 4221 . . . . . . . . . . . . 13 ((ran 𝑓 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝑓)
3432, 33sylibr 224 . . . . . . . . . . . 12 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → (ran 𝑓 ∩ {𝑥}) = ∅)
3534adantl 482 . . . . . . . . . . 11 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → (ran 𝑓 ∩ {𝑥}) = ∅)
3610adantr 481 . . . . . . . . . . . 12 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → ran 𝑓𝐵)
37 reldisj 3997 . . . . . . . . . . . 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 6066 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵 ∧ ran 𝑓 ⊆ (𝐵 ∖ {𝑥})) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4139, 40syldan 487 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓)) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
4241adantl 482 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑓:𝐴1-1→(𝐵 ∖ {𝑥}))
43 f1dom2g 7918 . . . . . . . 8 ((𝐴 ∈ V ∧ (𝐵 ∖ {𝑥}) ∈ V ∧ 𝑓:𝐴1-1→(𝐵 ∖ {𝑥})) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
4428, 31, 42, 43syl3anc 1323 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐴 ≼ (𝐵 ∖ {𝑥}))
45 eldifi 3715 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝑓) → 𝑥𝐵)
4645ad2antll 764 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝑥𝐵)
47 simplr 791 . . . . . . . 8 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → 𝐶𝐵)
48 difsnen 7987 . . . . . . . 8 ((𝐵 ∈ V ∧ 𝑥𝐵𝐶𝐵) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
4929, 46, 47, 48syl3anc 1323 . . . . . . 7 (((𝐴𝐵𝐶𝐵) ∧ (𝑓:𝐴1-1𝐵𝑥 ∈ (𝐵 ∖ ran 𝑓))) → (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶}))
50 domentr 7960 . . . . . . 7 ((𝐴 ≼ (𝐵 ∖ {𝑥}) ∧ (𝐵 ∖ {𝑥}) ≈ (𝐵 ∖ {𝐶})) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
5144, 49, 50syl2anc 692 . . . . . 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 1863 . . . 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 1865 . 2 ((𝐴𝐵𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
561adantr 481 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴𝐵)
57 difsn 4302 . . . . 5 𝐶𝐵 → (𝐵 ∖ {𝐶}) = 𝐵)
5857breq2d 4630 . . . 4 𝐶𝐵 → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
5958adantl 482 . . 3 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → (𝐴 ≼ (𝐵 ∖ {𝐶}) ↔ 𝐴𝐵))
6056, 59mpbird 247 . 2 ((𝐴𝐵 ∧ ¬ 𝐶𝐵) → 𝐴 ≼ (𝐵 ∖ {𝐶}))
6155, 60pm2.61dan 831 1 (𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  wne 2796  Vcvv 3191  cdif 3557  cin 3559  wss 3560  c0 3896  {csn 4153   class class class wbr 4618  ran crn 5080  wf 5846  1-1wf1 5847  1-1-ontowf1o 5849  cen 7897  cdom 7898  csdm 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-suc 5691  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903
This theorem is referenced by:  domunsn  8055  marypha1lem  8284
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