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Theorem brdom4 9635
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 9633 . . 3 (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
3 mormo 3345 . . . . . . 7 (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦𝐴 𝑥𝑓𝑦)
43alimi 1896 . . . . . 6 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦)
5 alral 3114 . . . . . 6 (∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
64, 5syl 17 . . . . 5 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
76anim1i 604 . . . 4 ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
87eximi 1919 . . 3 (∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
92, 8sylbi 208 . 2 (𝐴𝐵 → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
10 inss2 4028 . . . . . . . . . . . . . . 15 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11 dmss 5522 . . . . . . . . . . . . . . 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 5774 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐴) ⊆ 𝐵
1412, 13sstri 3805 . . . . . . . . . . . . 13 dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
1514sseli 3792 . . . . . . . . . . . 12 (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥𝐵)
16 rnss 5553 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴) → ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴))
1710, 16ax-mp 5 . . . . . . . . . . . . . . . . 17 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
18 rnxpss 5775 . . . . . . . . . . . . . . . . 17 ran (𝐵 × 𝐴) ⊆ 𝐴
1917, 18sstri 3805 . . . . . . . . . . . . . . . 16 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
2019sseli 3792 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦𝐴)
21 inss1 4027 . . . . . . . . . . . . . . . 16 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
2221ssbri 4887 . . . . . . . . . . . . . . 15 (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦𝑥𝑓𝑦)
2320, 22anim12i 602 . . . . . . . . . . . . . 14 ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦𝐴𝑥𝑓𝑦))
2423moimi 2681 . . . . . . . . . . . . 13 (∃*𝑦(𝑦𝐴𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
25 df-rmo 3102 . . . . . . . . . . . . 13 (∃*𝑦𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦𝐴𝑥𝑓𝑦))
26 df-rmo 3102 . . . . . . . . . . . . 13 (∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 ↔ ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
2724, 25, 263imtr4i 283 . . . . . . . . . . . 12 (∃*𝑦𝐴 𝑥𝑓𝑦 → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
2815, 27imim12i 62 . . . . . . . . . . 11 ((𝑥𝐵 → ∃*𝑦𝐴 𝑥𝑓𝑦) → (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
2928ralimi2 3135 . . . . . . . . . 10 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
30 relxp 5326 . . . . . . . . . . 11 Rel (𝐵 × 𝐴)
31 relin2 5436 . . . . . . . . . . 11 (Rel (𝐵 × 𝐴) → Rel (𝑓 ∩ (𝐵 × 𝐴)))
3230, 31ax-mp 5 . . . . . . . . . 10 Rel (𝑓 ∩ (𝐵 × 𝐴))
3329, 32jctil 511 . . . . . . . . 9 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
34 dffun9 6128 . . . . . . . . 9 (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
3533, 34sylibr 225 . . . . . . . 8 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → Fun (𝑓 ∩ (𝐵 × 𝐴)))
36 funfn 6129 . . . . . . . 8 (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
3735, 36sylib 209 . . . . . . 7 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
38 rninxp 5782 . . . . . . . 8 (ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥)
3938biimpri 219 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴)
4037, 39anim12i 602 . . . . . 6 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
41 df-fo 6105 . . . . . 6 ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴 ↔ ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
4240, 41sylibr 225 . . . . 5 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴)
43 vex 3392 . . . . . . . 8 𝑓 ∈ V
4443inex1 4992 . . . . . . 7 (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4544dmex 7327 . . . . . 6 dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4645fodom 9627 . . . . 5 ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
4742, 46syl 17 . . . 4 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
48 ssdomg 8236 . . . . 5 (𝐵 ∈ V → (dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵 → dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵))
491, 14, 48mp2 9 . . . 4 dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵
50 domtr 8243 . . . 4 ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴𝐵)
5147, 49, 50sylancl 576 . . 3 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
5251exlimiv 2021 . 2 (∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
539, 52impbii 200 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃*wmo 2631  wral 3094  wrex 3095  ∃*wrmo 3097  Vcvv 3389  cin 3766  wss 3767   class class class wbr 4842   × cxp 5307  dom cdm 5309  ran crn 5310  Rel wrel 5314  Fun wfun 6093   Fn wfn 6094  ontowfo 6097  cdom 8188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-ac2 9568
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-1st 7396  df-2nd 7397  df-wrecs 7640  df-recs 7702  df-er 7977  df-map 8092  df-en 8191  df-dom 8192  df-sdom 8193  df-card 9046  df-acn 9049  df-ac 9220
This theorem is referenced by:  brdom7disj  9636
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