Theorem fncnv 5386

Description: Single-rootedness (see funcnv 5381) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Assertion

Ref	Expression
fncnv	⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

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

Proof of Theorem fncnv

Step	Hyp	Ref	Expression
1		df-fn 5320	. 2 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2		df-rn 4729	. . . 4 ⊢ ran (𝑅 ∩ (𝐴 × 𝐵)) = dom ◡(𝑅 ∩ (𝐴 × 𝐵))
3	2	eqeq1i 2237	. . 3 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
4	3	anbi2i 457	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5		rninxp 5171	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦)
6	5	anbi1i 458	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
7		funcnv 5381	. . . . . 6 ⊢ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8		raleq 2728	. . . . . . 7 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦))
9		moanimv 2153	. . . . . . . . . 10 ⊢ (∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
10		brinxp2 4785	. . . . . . . . . . . 12 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
11		3anan12 1014	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
12	10, 11	bitri 184	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
13	12	mobii 2114	. . . . . . . . . 10 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
14		df-rmo 2516	. . . . . . . . . . 11 ⊢ (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
15	14	imbi2i 226	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
16	9, 13, 15	3bitr4i 212	. . . . . . . . 9 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
17		biimt 241	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)))
18	16, 17	bitr4id 199	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
19	18	ralbiia 2544	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)
20	8, 19	bitrdi 196	. . . . . 6 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
21	7, 20	bitrid 192	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
22	21	pm5.32i 454	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
23		r19.26 2657	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
24	6, 22, 23	3bitr4i 212	. . 3 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
25		ancom 266	. . 3 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))))
26		reu5 2749	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
27	26	ralbii 2536	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
28	24, 25, 27	3bitr4i 212	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
29	1, 4, 28	3bitr2i 208	1 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃*wmo 2078   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  ∃*wrmo 2511   ∩ cin 3196   class class class wbr 4082   × cxp 4716  ^◡ccnv 4717  dom cdm 4718  ran crn 4719  Fun wfun 5311   Fn wfn 5312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-fun 5319  df-fn 5320

This theorem is referenced by: (None)