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Theorem brdom4 9756
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 9754 . . 3 (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
3 mormo 3371 . . . . . . 7 (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦𝐴 𝑥𝑓𝑦)
43alimi 1775 . . . . . 6 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦)
5 alral 3106 . . . . . 6 (∀𝑥∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
64, 5syl 17 . . . . 5 (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦)
76anim1i 606 . . . 4 ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
87eximi 1798 . . 3 (∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
92, 8sylbi 209 . 2 (𝐴𝐵 → ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
10 inss2 4096 . . . . . . . . . . . . . . 15 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11 dmss 5625 . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐴) ⊆ 𝐵
1412, 13sstri 3869 . . . . . . . . . . . . 13 dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
1514sseli 3856 . . . . . . . . . . . 12 (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥𝐵)
1610rnssi 5658 . . . . . . . . . . . . . . . . 17 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
17 rnxpss 5874 . . . . . . . . . . . . . . . . 17 ran (𝐵 × 𝐴) ⊆ 𝐴
1816, 17sstri 3869 . . . . . . . . . . . . . . . 16 ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
1918sseli 3856 . . . . . . . . . . . . . . 15 (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦𝐴)
20 inss1 4095 . . . . . . . . . . . . . . . 16 (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
2120ssbri 4979 . . . . . . . . . . . . . . 15 (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦𝑥𝑓𝑦)
2219, 21anim12i 604 . . . . . . . . . . . . . 14 ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦𝐴𝑥𝑓𝑦))
2322moimi 2557 . . . . . . . . . . . . 13 (∃*𝑦(𝑦𝐴𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
24 df-rmo 3098 . . . . . . . . . . . . 13 (∃*𝑦𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦𝐴𝑥𝑓𝑦))
25 df-rmo 3098 . . . . . . . . . . . . 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 284 . . . . . . . . . . . 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 3109 . . . . . . . . . 10 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
29 relinxp 5541 . . . . . . . . . 10 Rel (𝑓 ∩ (𝐵 × 𝐴))
3028, 29jctil 512 . . . . . . . . 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 6222 . . . . . . . . 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 226 . . . . . . . 8 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → Fun (𝑓 ∩ (𝐵 × 𝐴)))
3332funfnd 6224 . . . . . . 7 (∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
34 rninxp 5881 . . . . . . . 8 (ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥)
3534biimpri 220 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴)
3633, 35anim12i 604 . . . . . 6 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
37 df-fo 6199 . . . . . 6 ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴 ↔ ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
3836, 37sylibr 226 . . . . 5 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → (𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto𝐴)
39 vex 3420 . . . . . . . 8 𝑓 ∈ V
4039inex1 5082 . . . . . . 7 (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4140dmex 7437 . . . . . 6 dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
4241fodom 9748 . . . . 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 8358 . . . . 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 8365 . . . 4 ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴𝐵)
4743, 45, 46sylancl 578 . . 3 ((∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
4847exlimiv 1890 . 2 (∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥) → 𝐴𝐵)
499, 48impbii 201 1 (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wal 1506   = wceq 1508  wex 1743  wcel 2051  ∃*wmo 2549  wral 3090  wrex 3091  ∃*wrmo 3093  Vcvv 3417  cin 3830  wss 3831   class class class wbr 4934   × cxp 5409  dom cdm 5411  ran crn 5412  Rel wrel 5416  Fun wfun 6187   Fn wfn 6188  ontowfo 6191  cdom 8310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-ac2 9689
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-se 5371  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-isom 6202  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-er 8095  df-map 8214  df-en 8313  df-dom 8314  df-sdom 8315  df-card 9168  df-acn 9171  df-ac 9342
This theorem is referenced by:  brdom7disj  9757
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