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Theorem mapdhval0 41107
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)
65fvexi 6898 . . . 4 0 ∈ V
76a1i 11 . . 3 (𝜑0 ∈ V)
81, 2, 3, 4, 7mapdhval 41106 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))))
9 eqid 2726 . . 3 0 = 0
109iftruei 4530 . 2 if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))) = 𝑄
118, 10eqtrdi 2782 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  ifcif 4523  {csn 4623  cotp 4631  cmpt 5224  cfv 6536  crio 7359  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  0gc0g 17392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-ov 7407  df-1st 7971  df-2nd 7972
This theorem is referenced by:  mapdhcl  41109  mapdh6bN  41119  mapdh6cN  41120  mapdh6dN  41121  mapdh8  41170
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