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Theorem mapdhval0 37879
 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 6460 . . . 4 0 ∈ V
76a1i 11 . . 3 (𝜑0 ∈ V)
81, 2, 3, 4, 7mapdhval 37878 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))))
9 eqid 2778 . . 3 0 = 0
109iftruei 4314 . 2 if( 0 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{ 0 })) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 0 )})) = (𝐽‘{(𝐹𝑅)})))) = 𝑄
118, 10syl6eq 2830 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ifcif 4307  {csn 4398  ⟨cotp 4406   ↦ cmpt 4965  ‘cfv 6135  ℩crio 6882  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  0gc0g 16486 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-ot 4407  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-riota 6883  df-ov 6925  df-1st 7445  df-2nd 7446 This theorem is referenced by:  mapdhcl  37881  mapdh6bN  37891  mapdh6cN  37892  mapdh6dN  37893  mapdh8  37942
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