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Theorem fncnv 6632
Description: Single-rootedness (see funcnv 6628) 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 6557 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2 df-rn 5693 . . . 4 ran (𝑅 ∩ (𝐴 × 𝐵)) = dom (𝑅 ∩ (𝐴 × 𝐵))
32eqeq1i 2731 . . 3 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
43anbi2i 621 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5 rninxp 6190 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦)
65anbi1i 622 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
7 funcnv 6628 . . . . . 6 (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8 raleq 3312 . . . . . . 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 moanimv 2608 . . . . . . . . . 10 (∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
10 brinxp2 5759 . . . . . . . . . . . 12 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
11 an21 642 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1210, 11bitri 274 . . . . . . . . . . 11 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1312mobii 2537 . . . . . . . . . 10 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
14 df-rmo 3364 . . . . . . . . . . 11 (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥𝐴𝑥𝑅𝑦))
1514imbi2i 335 . . . . . . . . . 10 ((𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
169, 13, 153bitr4i 302 . . . . . . . . 9 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦))
17 biimt 359 . . . . . . . . 9 (𝑦𝐵 → (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦)))
1816, 17bitr4id 289 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥𝐴 𝑥𝑅𝑦))
1918ralbiia 3081 . . . . . . 7 (∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦)
208, 19bitrdi 286 . . . . . 6 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
217, 20bitrid 282 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
2221pm5.32i 573 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
23 r19.26 3101 . . . 4 (∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
246, 22, 233bitr4i 302 . . 3 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
25 ancom 459 . . 3 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))))
26 reu5 3366 . . . 4 (∃!𝑥𝐴 𝑥𝑅𝑦 ↔ (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2726ralbii 3083 . . 3 (∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2824, 25, 273bitr4i 302 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
291, 4, 283bitr2i 298 1 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  ∃*wmo 2527  wral 3051  wrex 3060  ∃!wreu 3362  ∃*wrmo 3363  cin 3946   class class class wbr 5153   × cxp 5680  ccnv 5681  dom cdm 5682  ran crn 5683  Fun wfun 6548   Fn wfn 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6556  df-fn 6557
This theorem is referenced by: (None)
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