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Theorem fncnv 6207
Description: Single-rootedness (see funcnv 6203) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.)
Assertion
Ref Expression
fncnv ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem fncnv
StepHypRef Expression
1 df-fn 6138 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2 df-rn 5366 . . . 4 ran (𝑅 ∩ (𝐴 × 𝐵)) = dom (𝑅 ∩ (𝐴 × 𝐵))
32eqeq1i 2782 . . 3 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
43anbi2i 616 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5 rninxp 5827 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦)
65anbi1i 617 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
7 funcnv 6203 . . . . . 6 (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8 raleq 3329 . . . . . . 7 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦))
9 biimt 352 . . . . . . . . 9 (𝑦𝐵 → (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦)))
10 moanimv 2653 . . . . . . . . . 10 (∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
11 brinxp2 5426 . . . . . . . . . . . 12 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
12 an21 634 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1311, 12bitri 267 . . . . . . . . . . 11 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1413mobii 2562 . . . . . . . . . 10 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
15 df-rmo 3097 . . . . . . . . . . 11 (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥𝐴𝑥𝑅𝑦))
1615imbi2i 328 . . . . . . . . . 10 ((𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
1710, 14, 163bitr4i 295 . . . . . . . . 9 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦))
189, 17syl6rbbr 282 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥𝐴 𝑥𝑅𝑦))
1918ralbiia 3160 . . . . . . 7 (∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦)
208, 19syl6bb 279 . . . . . 6 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
217, 20syl5bb 275 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
2221pm5.32i 570 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
23 r19.26 3249 . . . 4 (∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
246, 22, 233bitr4i 295 . . 3 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
25 ancom 454 . . 3 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))))
26 reu5 3354 . . . 4 (∃!𝑥𝐴 𝑥𝑅𝑦 ↔ (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2726ralbii 3161 . . 3 (∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2824, 25, 273bitr4i 295 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
291, 4, 283bitr2i 291 1 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  ∃*wmo 2548  wral 3089  wrex 3090  ∃!wreu 3091  ∃*wrmo 3092  cin 3790   class class class wbr 4886   × cxp 5353  ccnv 5354  dom cdm 5355  ran crn 5356  Fun wfun 6129   Fn wfn 6130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-fun 6137  df-fn 6138
This theorem is referenced by: (None)
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