Theorem brdom4 9352

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 9350	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
3		mormo 3158	. . . . . . 7 ⊢ (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
4	3	alimi 1739	. . . . . 6 ⊢ (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
5		alral 2928	. . . . . 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 592	. . . 4 ⊢ ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
8	7	eximi 1762	. . 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 207	. 2 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
10		inss2 3834	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11		dmss 5323	. . . . . . . . . . . . . . 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 5565	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
14	12, 13	sstri 3612	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
15	14	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥 ∈ 𝐵)
16		rnss 5354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴) → ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴))
17	10, 16	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
18		rnxpss 5566	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝐵 × 𝐴) ⊆ 𝐴
19	17, 18	sstri 3612	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
20	19	sseli 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦 ∈ 𝐴)
21		inss1 3833	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
22	21	ssbri 4697	. . . . . . . . . . . . . . 15 ⊢ (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 → 𝑥𝑓𝑦)
23	20, 22	anim12i 590	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
24	23	moimi 2520	. . . . . . . . . . . . 13 ⊢ (∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
25		df-rmo 2920	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
26		df-rmo 2920	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 ↔ ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
27	24, 25, 26	3imtr4i 281	. . . . . . . . . . . 12 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
28	15, 27	imim12i 62	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦) → (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → ∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
29	28	ralimi2 2949	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
30		relxp 5227	. . . . . . . . . . 11 ⊢ Rel (𝐵 × 𝐴)
31		relin2 5237	. . . . . . . . . . 11 ⊢ (Rel (𝐵 × 𝐴) → Rel (𝑓 ∩ (𝐵 × 𝐴)))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ Rel (𝑓 ∩ (𝐵 × 𝐴))
33	29, 32	jctil 560	. . . . . . . . 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))
34		dffun9 5917	. . . . . . . . 9 ⊢ (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (Rel (𝑓 ∩ (𝐵 × 𝐴)) ∧ ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
35	33, 34	sylibr 224	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → Fun (𝑓 ∩ (𝐵 × 𝐴)))
36		funfn 5918	. . . . . . . 8 ⊢ (Fun (𝑓 ∩ (𝐵 × 𝐴)) ↔ (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
37	35, 36	sylib 208	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
38		rninxp 5573	. . . . . . . 8 ⊢ (ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)
39	38	biimpri 218	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴)
40	37, 39	anim12i 590	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
41		df-fo 5894	. . . . . 6 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴 ↔ ((𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ ran (𝑓 ∩ (𝐵 × 𝐴)) = 𝐴))
42	40, 41	sylibr 224	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → (𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴)
43		vex 3203	. . . . . . . 8 ⊢ 𝑓 ∈ V
44	43	inex1 4799	. . . . . . 7 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
45	44	dmex 7099	. . . . . 6 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
46	45	fodom 9344	. . . . 5 ⊢ ((𝑓 ∩ (𝐵 × 𝐴)):dom (𝑓 ∩ (𝐵 × 𝐴))–onto→𝐴 → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
47	42, 46	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)))
48		ssdomg 8001	. . . . 5 ⊢ (𝐵 ∈ V → (dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵 → dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵))
49	1, 14, 48	mp2 9	. . . 4 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵
50		domtr 8009	. . . 4 ⊢ ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴 ≼ 𝐵)
51	47, 49, 50	sylancl 694	. . 3 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ 𝐵)
52	51	exlimiv 1858	. 2 ⊢ (∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ 𝐵)
53	9, 52	impbii 199	1 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃*wmo 2471  ∀wral 2912  ∃wrex 2913  ∃*wrmo 2915  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653   × cxp 5112  dom cdm 5114  ran crn 5115  Rel wrel 5119  Fun wfun 5882   Fn wfn 5883  –onto→wfo 5886   ≼ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by:  brdom7disj  9353