Theorem brdom4 10542

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	⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem brdom4

Step	Hyp	Ref	Expression
1		brdom3.2	. . . 4 ⊢ 𝐵 ∈ V
2	1	brdom3 10540	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
3		mormo 3364	. . . . . . 7 ⊢ (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
4	3	alimi 1811	. . . . . 6 ⊢ (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
5		alral 3065	. . . . . 6 ⊢ (∀𝑥∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
6	4, 5	syl 17	. . . . 5 ⊢ (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
7	6	anim1i 615	. . . 4 ⊢ ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
8	7	eximi 1835	. . 3 ⊢ (∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
9	2, 8	sylbi 217	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
10		inss2 4213	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11		dmss 5882	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴) → dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ dom (𝐵 × 𝐴))
12	10, 11	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ dom (𝐵 × 𝐴)
13		dmxpss 6160	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
14	12, 13	sstri 3968	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
15	14	sseli 3954	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥 ∈ 𝐵)
16	10	rnssi 5920	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
17		rnxpss 6161	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝐵 × 𝐴) ⊆ 𝐴
18	16, 17	sstri 3968	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
19	18	sseli 3954	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦 ∈ 𝐴)
20		inss1 4212	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
21	20	ssbri 5164	. . . . . . . . . . . . . . 15 ⊢ (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 → 𝑥𝑓𝑦)
22	19, 21	anim12i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
23	22	moimi 2544	. . . . . . . . . . . . 13 ⊢ (∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
24		df-rmo 3359	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
25		df-rmo 3359	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 ↔ ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
26	23, 24, 25	3imtr4i 292	. . . . . . . . . . . 12 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
27	15, 26	imim12i 62	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦) → (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
28	27	ralimi2 3068	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
29		relinxp 5793	. . . . . . . . . 10 ⊢ Rel (𝑓 ∩ (𝐵 × 𝐴))
30	28, 29	jctil 519	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
31		dffun9 6564	. . . . . . . . 9 ⊢ (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
32	30, 31	sylibr 234	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → Fun (𝑓 ∩ (𝐵 × 𝐴)))
33	32	funfnd 6566	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
34		rninxp 6168	. . . . . . . 8 ⊢ (ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)
35	34	biimpri 228	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴)
36	33, 35	anim12i 613	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
37		df-fo 6536	. . . . . 6 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴 ↔ ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
38	36, 37	sylibr 234	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → (𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴)
39		vex 3463	. . . . . . . 8 ⊢ 𝑓 ∈ V
40	39	inex1 5287	. . . . . . 7 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
41	40	dmex 7903	. . . . . 6 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
42	41	fodom 10535	. . . . 5 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴 → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
43	38, 42	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
44		ssdomg 9012	. . . . 5 ⊢ (𝐵 ∈ V → (dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵 → dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵))
45	1, 14, 44	mp2 9	. . . 4 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵
46		domtr 9019	. . . 4 ⊢ ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴 ≼ 𝐵)
47	43, 45, 46	sylancl 586	. . 3 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ 𝐵)
48	47	exlimiv 1930	. 2 ⊢ (∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ 𝐵)
49	9, 48	impbii 209	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃*wmo 2537  ∀wral 3051  ∃wrex 3060  ∃*wrmo 3358  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926   class class class wbr 5119   × cxp 5652  dom cdm 5654  ran crn 5655  Rel wrel 5659  Fun wfun 6524   Fn wfn 6525  –onto→wfo 6528   ≼ cdom 8955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-ac2 10475

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-er 8717  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-card 9951  df-acn 9954  df-ac 10128

This theorem is referenced by:  brdom7disj  10543