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Theorem mapdhval0 40178
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 6856 . . . 4 0 ∈ V
76a1i 11 . . 3 (𝜑0 ∈ V)
81, 2, 3, 4, 7mapdhval 40177 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))))
9 eqid 2736 . . 3 0 = 0
109iftruei 4493 . 2 if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))) = 𝑄
118, 10eqtrdi 2792 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  ifcif 4486  {csn 4586  cotp 4594  cmpt 5188  cfv 6496  crio 7311  (class class class)co 7356  1st c1st 7918  2nd c2nd 7919  0gc0g 17320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fv 6504  df-riota 7312  df-ov 7359  df-1st 7920  df-2nd 7921
This theorem is referenced by:  mapdhcl  40180  mapdh6bN  40190  mapdh6cN  40191  mapdh6dN  40192  mapdh8  40241
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