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Theorem mapdhval0 36032
 Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
Hypotheses
Ref Expression
mapdh.q 𝑄 = (0g𝐶)
mapdh.i 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
mapdh0.o 0 = (0g𝑈)
mapdh0.x (𝜑𝑋𝐴)
mapdh0.f (𝜑𝐹𝐵)
Assertion
Ref Expression
mapdhval0 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
Distinct variable groups:   𝑥,𝐷   𝑥,,𝐹   𝑥,𝐽   𝑥,𝑀   𝑥,𝑁   𝑥, 0   𝑥,𝑄   𝑥,𝑅   𝑥,   ,𝑋,𝑥   𝜑,   0 ,
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,)   𝐵(𝑥,)   𝐶(𝑥,)   𝐷()   𝑄()   𝑅()   𝑈(𝑥,)   𝐼(𝑥,)   𝐽()   𝑀()   ()   𝑁()

Proof of Theorem mapdhval0
StepHypRef Expression
1 mapdh.q . . 3 𝑄 = (0g𝐶)
2 mapdh.i . . 3 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
3 mapdh0.x . . 3 (𝜑𝑋𝐴)
4 mapdh0.f . . 3 (𝜑𝐹𝐵)
5 mapdh0.o . . . . 5 0 = (0g𝑈)
6 fvex 6113 . . . . 5 (0g𝑈) ∈ V
75, 6eqeltri 2684 . . . 4 0 ∈ V
87a1i 11 . . 3 (𝜑0 ∈ V)
91, 2, 3, 4, 8mapdhval 36031 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))))
10 eqid 2610 . . 3 0 = 0
1110iftruei 4043 . 2 if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))) = 𝑄
129, 11syl6eq 2660 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036  {csn 4125  ⟨cotp 4133   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0gc0g 15923 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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-ot 4134  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060 This theorem is referenced by:  mapdhcl  36034  mapdh6bN  36044  mapdh6cN  36045  mapdh6dN  36046  mapdh8  36096
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