Theorem fncnv 6397

Description: Single-rootedness (see funcnv 6393) of a class cut down by a Cartesian 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 6327	. 2 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2		df-rn 5530	. . . 4 ⊢ ran (𝑅 ∩ (𝐴 × 𝐵)) = dom ◡(𝑅 ∩ (𝐴 × 𝐵))
3	2	eqeq1i 2803	. . 3 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
4	3	anbi2i 625	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5		rninxp 6003	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦)
6	5	anbi1i 626	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
7		funcnv 6393	. . . . . 6 ⊢ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8		raleq 3358	. . . . . . 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 2681	. . . . . . . . . 10 ⊢ (∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
10		brinxp2 5593	. . . . . . . . . . . 12 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
11		an21 643	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
12	10, 11	bitri 278	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
13	12	mobii 2606	. . . . . . . . . 10 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
14		df-rmo 3114	. . . . . . . . . . 11 ⊢ (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
15	14	imbi2i 339	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
16	9, 13, 15	3bitr4i 306	. . . . . . . . 9 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
17		biimt 364	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)))
18	16, 17	bitr4id 293	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
19	18	ralbiia 3132	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)
20	8, 19	syl6bb 290	. . . . . 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	syl5bb 286	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
22	21	pm5.32i 578	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
23		r19.26 3137	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
24	6, 22, 23	3bitr4i 306	. . 3 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
25		ancom 464	. . 3 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))))
26		reu5 3375	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
27	26	ralbii 3133	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
28	24, 25, 27	3bitr4i 306	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
29	1, 4, 28	3bitr2i 302	1 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  ∃*wrmo 3109   ∩ cin 3880   class class class wbr 5030   × cxp 5517  ^◡ccnv 5518  dom cdm 5519  ran crn 5520  Fun wfun 6318   Fn wfn 6319

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326  df-fn 6327

This theorem is referenced by: (None)