Theorem fncnv 6422

Description: Single-rootedness (see funcnv 6418) 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 6353	. 2 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2		df-rn 5561	. . . 4 ⊢ ran (𝑅 ∩ (𝐴 × 𝐵)) = dom ◡(𝑅 ∩ (𝐴 × 𝐵))
3	2	eqeq1i 2826	. . 3 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
4	3	anbi2i 624	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5		rninxp 6031	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦)
6	5	anbi1i 625	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
7		funcnv 6418	. . . . . 6 ⊢ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8		raleq 3406	. . . . . . 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		biimt 363	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)))
10		moanimv 2700	. . . . . . . . . 10 ⊢ (∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
11		brinxp2 5624	. . . . . . . . . . . 12 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦))
12		an21 642	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
13	11, 12	bitri 277	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
14	13	mobii 2627	. . . . . . . . . 10 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
15		df-rmo 3146	. . . . . . . . . . 11 ⊢ (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
16	15	imbi2i 338	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
17	10, 14, 16	3bitr4i 305	. . . . . . . . 9 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
18	9, 17	syl6rbbr 292	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
19	18	ralbiia 3164	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)
20	8, 19	syl6bb 289	. . . . . 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 285	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
22	21	pm5.32i 577	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
23		r19.26 3170	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
24	6, 22, 23	3bitr4i 305	. . 3 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
25		ancom 463	. . 3 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))))
26		reu5 3431	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
27	26	ralbii 3165	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
28	24, 25, 27	3bitr4i 305	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
29	1, 4, 28	3bitr2i 301	1 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃*wmo 2616  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  ∃*wrmo 3141   ∩ cin 3935   class class class wbr 5059   × cxp 5548  ^◡ccnv 5549  dom cdm 5550  ran crn 5551  Fun wfun 6344   Fn wfn 6345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-fun 6352  df-fn 6353

This theorem is referenced by: (None)