Theorem brdom4 10452

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 10450	. . 3 ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
3		mormo 3357	. . . . . . 7 ⊢ (∃*𝑦 𝑥𝑓𝑦 → ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
4	3	alimi 1813	. . . . . 6 ⊢ (∀𝑥∃*𝑦 𝑥𝑓𝑦 → ∀𝑥∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦)
5		alral 3067	. . . . . 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 616	. . . 4 ⊢ ((∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥))
8	7	eximi 1837	. . 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 4192	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ (𝐵 × 𝐴)
11		dmss 5859	. . . . . . . . . . . . . . 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 6137	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐴) ⊆ 𝐵
14	12, 13	sstri 3945	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐵
15	14	sseli 3931	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴)) → 𝑥 ∈ 𝐵)
16	10	rnssi 5897	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ ran (𝐵 × 𝐴)
17		rnxpss 6138	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝐵 × 𝐴) ⊆ 𝐴
18	16, 17	sstri 3945	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝐴
19	18	sseli 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) → 𝑦 ∈ 𝐴)
20		inss1 4191	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ⊆ 𝑓
21	20	ssbri 5145	. . . . . . . . . . . . . . 15 ⊢ (𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦 → 𝑥𝑓𝑦)
22	19, 21	anim12i 614	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦) → (𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
23	22	moimi 2546	. . . . . . . . . . . . 13 ⊢ (∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦) → ∃*𝑦(𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴)) ∧ 𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦))
24		df-rmo 3352	. . . . . . . . . . . . 13 ⊢ (∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑥𝑓𝑦))
25		df-rmo 3352	. . . . . . . . . . . . 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 3070	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → ∀𝑥 ∈ dom (𝑓 ∩ (𝐵 × 𝐴))∃*𝑦 ∈ ran (𝑓 ∩ (𝐵 × 𝐴))𝑥(𝑓 ∩ (𝐵 × 𝐴))𝑦)
29		relinxp 5771	. . . . . . . . . 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 6529	. . . . . . . . 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 6531	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 → (𝑓 ∩ (𝐵 × 𝐴)) Fn dom (𝑓 ∩ (𝐵 × 𝐴)))
34		rninxp 6145	. . . . . . . 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 614	. . . . . 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 6506	. . . . . 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 3446	. . . . . . . 8 ⊢ 𝑓 ∈ V
40	39	inex1 5264	. . . . . . 7 ⊢ (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
41	40	dmex 7861	. . . . . 6 ⊢ dom (𝑓 ∩ (𝐵 × 𝐴)) ∈ V
42	41	fodom 10445	. . . . 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 8949	. . . . 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 8956	. . . 4 ⊢ ((𝐴 ≼ dom (𝑓 ∩ (𝐵 × 𝐴)) ∧ dom (𝑓 ∩ (𝐵 × 𝐴)) ≼ 𝐵) → 𝐴 ≼ 𝐵)
47	43, 45, 46	sylancl 587	. . 3 ⊢ ((∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥) → 𝐴 ≼ 𝐵)
48	47	exlimiv 1932	. 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  ∃wrex 3062  ∃*wrmo 3351  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903   class class class wbr 5100   × cxp 5630  dom cdm 5632  ran crn 5633  Rel wrel 5637  Fun wfun 6494   Fn wfn 6495  –onto→wfo 6498   ≼ cdom 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-card 9863  df-acn 9866  df-ac 10038

This theorem is referenced by:  brdom7disj  10453