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Theorem brdom4 9940
Description: An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
Hypothesis
Ref Expression
brdom3.2 𝐵 ∈ V
Assertion
Ref Expression
brdom4 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem brdom4
StepHypRef Expression
1 brdom3.2 . . . 4 𝐵 ∈ V
21brdom3 9938 . . 3 (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
3 mormo 3427 . . . . . . 7 (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦𝐴 𝑥𝑓𝑦)
43alimi 1803 . . . . . 6 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦)
5 alral 3151 . . . . . 6 (∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
64, 5syl 17 . . . . 5 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
76anim1i 614 . . . 4 ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
87eximi 1826 . . 3 (∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
92, 8sylbi 218 . 2 (𝐴𝐵 → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
10 inss2 4203 . . . . . . . . . . . . . . 15 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11 dmss 5764 . . . . . . . . . . . . . . 15 ((𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴) → dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ dom (𝐵 × 𝐴))
1210, 11ax-mp 5 . . . . . . . . . . . . . 14 dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ dom (𝐵 × 𝐴)
13 dmxpss 6021 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐴) ⊆ 𝐵
1412, 13sstri 3973 . . . . . . . . . . . . 13 dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
1514sseli 3960 . . . . . . . . . . . 12 (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥𝐵)
1610rnssi 5803 . . . . . . . . . . . . . . . . 17 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
17 rnxpss 6022 . . . . . . . . . . . . . . . . 17 ran (𝐵 × 𝐴) ⊆ 𝐴
1816, 17sstri 3973 . . . . . . . . . . . . . . . 16 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
1918sseli 3960 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦𝐴)
20 inss1 4202 . . . . . . . . . . . . . . . 16 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
2120ssbri 5102 . . . . . . . . . . . . . . 15 (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦𝑥𝑓𝑦)
2219, 21anim12i 612 . . . . . . . . . . . . . 14 ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦𝐴𝑥𝑓𝑦))
2322moimi 2620 . . . . . . . . . . . . 13 (∃*𝑦(𝑦𝐴𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
24 df-rmo 3143 . . . . . . . . . . . . 13 (∃*𝑦𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦𝐴𝑥𝑓𝑦))
25 df-rmo 3143 . . . . . . . . . . . . 13 (∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 ↔ ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
2623, 24, 253imtr4i 293 . . . . . . . . . . . 12 (∃*𝑦𝐴 𝑥𝑓𝑦 → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
2715, 26imim12i 62 . . . . . . . . . . 11 ((𝑥𝐵 → ∃*𝑦𝐴 𝑥𝑓𝑦) → (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
2827ralimi2 3154 . . . . . . . . . 10 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
29 relinxp 5680 . . . . . . . . . 10 Rel (𝑓 ∩ (𝐵 × 𝐴))
3028, 29jctil 520 . . . . . . . . 9 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
31 dffun9 6377 . . . . . . . . 9 (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
3230, 31sylibr 235 . . . . . . . 8 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → Fun (𝑓 ∩ (𝐵 × 𝐴)))
3332funfnd 6379 . . . . . . 7 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
34 rninxp 6029 . . . . . . . 8 (ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥)
3534biimpri 229 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴)
3633, 35anim12i 612 . . . . . 6 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
37 df-fo 6354 . . . . . 6 ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴 ↔ ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
3836, 37sylibr 235 . . . . 5 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴)
39 vex 3495 . . . . . . . 8 𝑓 ∈ V
4039inex1 5212 . . . . . . 7 (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4140dmex 7605 . . . . . 6 dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4241fodom 9932 . . . . 5 ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
4338, 42syl 17 . . . 4 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
44 ssdomg 8543 . . . . 5 (𝐵 ∈ V → (dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵 → dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵))
451, 14, 44mp2 9 . . . 4 dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵
46 domtr 8550 . . . 4 ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴𝐵)
4743, 45, 46sylancl 586 . . 3 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
4847exlimiv 1922 . 2 (∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
499, 48impbii 210 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  wral 3135  wrex 3136  ∃*wrmo 3138  Vcvv 3492  cin 3932  wss 3933   class class class wbr 5057   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553  Fun wfun 6342   Fn wfn 6343  ontowfo 6346  cdom 8495
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-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  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-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  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-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-card 9356  df-acn 9359  df-ac 9530
This theorem is referenced by:  brdom7disj  9941
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