Theorem fncnv 6101

Description: Single-rootedness (see funcnv 6097) 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 6033	. 2 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2		df-rn 5261	. . . 4 ⊢ ran (𝑅 ∩ (𝐴 × 𝐵)) = dom ◡(𝑅 ∩ (𝐴 × 𝐵))
3	2	eqeq1i 2776	. . 3 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
4	3	anbi2i 609	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5		rninxp 5713	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦)
6	5	anbi1i 610	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
7		funcnv 6097	. . . . . 6 ⊢ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8		raleq 3287	. . . . . . 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 349	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)))
10		moanimv 2680	. . . . . . . . . 10 ⊢ (∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
11		brinxp2 5319	. . . . . . . . . . . 12 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
12		3anan12 1081	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
13	11, 12	bitri 264	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
14	13	mobii 2641	. . . . . . . . . 10 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
15		df-rmo 3069	. . . . . . . . . . 11 ⊢ (∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
16	15	imbi2i 325	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
17	10, 14, 16	3bitr4i 292	. . . . . . . . 9 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
18	9, 17	syl6rbbr 279	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
19	18	ralbiia 3128	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)
20	8, 19	syl6bb 276	. . . . . 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 272	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
22	21	pm5.32i 564	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
23		r19.26 3212	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
24	6, 22, 23	3bitr4i 292	. . 3 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
25		ancom 448	. . 3 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))))
26		reu5 3308	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
27	26	ralbii 3129	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
28	24, 25, 27	3bitr4i 292	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
29	1, 4, 28	3bitr2i 288	1 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∃*wmo 2619  ∀wral 3061  ∃wrex 3062  ∃!wreu 3063  ∃*wrmo 3064   ∩ cin 3722   class class class wbr 4787   × cxp 5248  ^◡ccnv 5249  dom cdm 5250  ran crn 5251  Fun wfun 6024   Fn wfn 6025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-fun 6032  df-fn 6033

This theorem is referenced by: (None)