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Theorem mapdhval0 36833
 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 6188 . . . . 5 (0g𝑈) ∈ V
75, 6eqeltri 2695 . . . 4 0 ∈ V
87a1i 11 . . 3 (𝜑0 ∈ V)
91, 2, 3, 4, 8mapdhval 36832 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))))
10 eqid 2620 . . 3 0 = 0
1110iftruei 4084 . 2 if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))) = 𝑄
129, 11syl6eq 2670 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ifcif 4077  {csn 4168  ⟨cotp 4176   ↦ cmpt 4720  ‘cfv 5876  ℩crio 6595  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  0gc0g 16081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-ot 4177  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-riota 6596  df-ov 6638  df-1st 7153  df-2nd 7154 This theorem is referenced by:  mapdhcl  36835  mapdh6bN  36845  mapdh6cN  36846  mapdh6dN  36847  mapdh8  36897
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