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Theorem fncnv 6619
Description: Single-rootedness (see funcnv 6615) 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 6544 . 2 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
2 df-rn 5687 . . . 4 ran (𝑅 ∩ (𝐴 × 𝐵)) = dom (𝑅 ∩ (𝐴 × 𝐵))
32eqeq1i 2738 . . 3 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵)
43anbi2i 624 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ dom (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵))
5 rninxp 6176 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦)
65anbi1i 625 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
7 funcnv 6615 . . . . . 6 (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦)
8 raleq 3323 . . . . . . 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 2616 . . . . . . . . . 10 (∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
10 brinxp2 5752 . . . . . . . . . . . 12 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦))
11 an21 643 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ 𝑥𝑅𝑦) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1210, 11bitri 275 . . . . . . . . . . 11 (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
1312mobii 2543 . . . . . . . . . 10 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵 ∧ (𝑥𝐴𝑥𝑅𝑦)))
14 df-rmo 3377 . . . . . . . . . . 11 (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ ∃*𝑥(𝑥𝐴𝑥𝑅𝑦))
1514imbi2i 336 . . . . . . . . . 10 ((𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (𝑦𝐵 → ∃*𝑥(𝑥𝐴𝑥𝑅𝑦)))
169, 13, 153bitr4i 303 . . . . . . . . 9 (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦))
17 biimt 361 . . . . . . . . 9 (𝑦𝐵 → (∃*𝑥𝐴 𝑥𝑅𝑦 ↔ (𝑦𝐵 → ∃*𝑥𝐴 𝑥𝑅𝑦)))
1816, 17bitr4id 290 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∃*𝑥𝐴 𝑥𝑅𝑦))
1918ralbiia 3092 . . . . . . 7 (∀𝑦𝐵 ∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦)
208, 19bitrdi 287 . . . . . 6 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (∀𝑦 ∈ ran (𝑅 ∩ (𝐴 × 𝐵))∃*𝑥 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
217, 20bitrid 283 . . . . 5 (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 → (Fun (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
2221pm5.32i 576 . . . 4 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
23 r19.26 3112 . . . 4 (∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦) ↔ (∀𝑦𝐵𝑥𝐴 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 ∃*𝑥𝐴 𝑥𝑅𝑦))
246, 22, 233bitr4i 303 . . 3 ((ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))) ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
25 ancom 462 . . 3 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ (ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵 ∧ Fun (𝑅 ∩ (𝐴 × 𝐵))))
26 reu5 3379 . . . 4 (∃!𝑥𝐴 𝑥𝑅𝑦 ↔ (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2726ralbii 3094 . . 3 (∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 (∃𝑥𝐴 𝑥𝑅𝑦 ∧ ∃*𝑥𝐴 𝑥𝑅𝑦))
2824, 25, 273bitr4i 303 . 2 ((Fun (𝑅 ∩ (𝐴 × 𝐵)) ∧ ran (𝑅 ∩ (𝐴 × 𝐵)) = 𝐵) ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
291, 4, 283bitr2i 299 1 ((𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ∃*wmo 2533  wral 3062  wrex 3071  ∃!wreu 3375  ∃*wrmo 3376  cin 3947   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  Fun wfun 6535   Fn wfn 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6543  df-fn 6544
This theorem is referenced by: (None)
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