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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 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem fncnv
StepHypRef 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))
32eqeq1i 2776 . . 3 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
43anbi2i 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 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦)
65anbi1i 610 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 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 ↔ ∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦))
9 biimt 349 . . . . . . . . 9 (𝑦𝐵 → (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦)))
10 moanimv 2680 . . . . . . . . . 10 (∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
11 brinxp2 5319 . . . . . . . . . . . 12 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑥𝐴𝑦𝐵𝑥𝑅𝑦))
12 3anan12 1081 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵𝑥𝑅𝑦) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1311, 12bitri 264 . . . . . . . . . . 11 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1413mobii 2641 . . . . . . . . . 10 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
15 df-rmo 3069 . . . . . . . . . . 11 (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥𝐴𝑥𝑅𝑦))
1615imbi2i 325 . . . . . . . . . 10 ((𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
1710, 14, 163bitr4i 292 . . . . . . . . 9 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦))
189, 17syl6rbbr 279 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥𝐴 𝑥𝑅𝑦))
1918ralbiia 3128 . . . . . . 7 (∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦)
208, 19syl6bb 276 . . . . . 6 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
217, 20syl5bb 272 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
2221pm5.32i 564 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
23 r19.26 3212 . . . 4 (∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
246, 22, 233bitr4i 292 . . 3 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 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 (∃!𝑥𝐴 𝑥𝑅𝑦 ↔ (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2726ralbii 3129 . . 3 (∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2824, 25, 273bitr4i 292 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
291, 4, 283bitr2i 288 1 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Colors of variables: wff setvar class
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)
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