Theorem fncnv 5876

Description: Single-rootedness (see funcnv 5872) 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 5807	. 2 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2		df-rn 5049	. . . 4 ⊢ ran (𝑅 ∩ (𝐴 × 𝐵)) = dom ◡(𝑅 ∩ (𝐴 × 𝐵))
3	2	eqeq1i 2615	. . 3 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
4	3	anbi2i 726	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ dom ◡(𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5		rninxp 5492	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦)
6	5	anbi1i 727	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
7		funcnv 5872	. . . . . 6 ⊢ (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8		raleq 3115	. . . . . . 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 2519	. . . . . . . . . 10 ⊢ (∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)) ↔ (𝑦 ∈ 𝐵 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
11		brinxp2 5103	. . . . . . . . . . . 12 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦))
12		3anan12 1044	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
13	11, 12	bitri 263	. . . . . . . . . . 11 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
14	13	mobii 2481	. . . . . . . . . 10 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)))
15		df-rmo 2904	. . . . . . . . . . 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 291	. . . . . . . . 9 ⊢ (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
18	9, 17	syl6rbbr 278	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
19	18	ralbiia 2962	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦)
20	8, 19	syl6bb 275	. . . . . 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 271	. . . . 5 ⊢ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
22	21	pm5.32i 667	. . . 4 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
23		r19.26 3046	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦) ↔ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
24	6, 22, 23	3bitr4i 291	. . 3 ⊢ ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
25		ancom 465	. . 3 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun ◡(𝑅 ∩ (𝐴 × 𝐵))))
26		reu5 3136	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
27	26	ralbii 2963	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 (∃𝑥 ∈ 𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥 ∈ 𝐴 𝑥𝑅𝑦))
28	24, 25, 27	3bitr4i 291	. 2 ⊢ ((Fun ◡(𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
29	1, 4, 28	3bitr2i 287	1 ⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899   ∩ cin 3539   class class class wbr 4583   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807

This theorem is referenced by: (None)