Theorem brdom4 10440

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 10438	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
3		mormo 3355	. . . . . . 7 ⊢ (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
4	3	alimi 1812	. . . . . 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 1836	. . 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 4190	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11		dmss 5851	. . . . . . . . . . . . . . 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 6129	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
14	12, 13	sstri 3943	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
15	14	sseli 3929	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥 ∈ 𝐵)
16	10	rnssi 5889	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
17		rnxpss 6130	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝐵 × 𝐴) ⊆ 𝐴
18	16, 17	sstri 3943	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
19	18	sseli 3929	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦 ∈ 𝐴)
20		inss1 4189	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
21	20	ssbri 5143	. . . . . . . . . . . . . . 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 2545	. . . . . . . . . . . . 13 ⊢ (∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
24		df-rmo 3350	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
25		df-rmo 3350	. . . . . . . . . . . . 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 5763	. . . . . . . . . 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 6521	. . . . . . . . 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 6523	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
34		rninxp 6137	. . . . . . . 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 6498	. . . . . 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 3444	. . . . . . . 8 ⊢ 𝑓 ∈ V
40	39	inex1 5262	. . . . . . 7 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
41	40	dmex 7851	. . . . . 6 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
42	41	fodom 10433	. . . . 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 8937	. . . . 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 8944	. . . 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 1931	. 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2537  ∀wral 3051  ∃wrex 3060  ∃*wrmo 3349  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   class class class wbr 5098   × cxp 5622  dom cdm 5624  ran crn 5625  Rel wrel 5629  Fun wfun 6486   Fn wfn 6487  –onto→wfo 6490   ≼ cdom 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-ac2 10373

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-card 9851  df-acn 9854  df-ac 10026

This theorem is referenced by:  brdom7disj  10441