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Theorem mapdhval 37745
Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-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)})))))
mapdh.x (𝜑𝑋𝐴)
mapdh.f (𝜑𝐹𝐵)
mapdh.y (𝜑𝑌𝐸)
Assertion
Ref Expression
mapdhval (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
Distinct variable groups:   𝑥,𝐷   𝑥,,𝐹   𝑥,𝐽   𝑥,𝑀   𝑥,𝑁   𝑥, 0   𝑥,𝑄   𝑥,𝑅   𝑥,   ,𝑋,𝑥   ,𝑌,𝑥   𝜑,
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,)   𝐵(𝑥,)   𝐶(𝑥,)   𝐷()   𝑄()   𝑅()   𝐸(𝑥,)   𝐼(𝑥,)   𝐽()   𝑀()   ()   𝑁()   0 ()

Proof of Theorem mapdhval
StepHypRef Expression
1 otex 5124 . . 3 𝑋, 𝐹, 𝑌⟩ ∈ V
2 fveq2 6411 . . . . . 6 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd𝑥) = (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))
32eqeq1d 2801 . . . . 5 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd𝑥) = 0 ↔ (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 ))
42sneqd 4380 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {(2nd𝑥)} = {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})
54fveq2d 6415 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{(2nd𝑥)}) = (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}))
65fveqeq2d 6419 . . . . . . 7 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{})))
7 fveq2 6411 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st𝑥) = (1st ‘⟨𝑋, 𝐹, 𝑌⟩))
87fveq2d 6415 . . . . . . . . . . . 12 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (1st ‘(1st𝑥)) = (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
98, 2oveq12d 6896 . . . . . . . . . . 11 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((1st ‘(1st𝑥)) (2nd𝑥)) = ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)))
109sneqd 4380 . . . . . . . . . 10 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((1st ‘(1st𝑥)) (2nd𝑥))} = {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})
1110fveq2d 6415 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}) = (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}))
1211fveq2d 6415 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})))
137fveq2d 6415 . . . . . . . . . . 11 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (2nd ‘(1st𝑥)) = (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)))
1413oveq1d 6893 . . . . . . . . . 10 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((2nd ‘(1st𝑥))𝑅) = ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅))
1514sneqd 4380 . . . . . . . . 9 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → {((2nd ‘(1st𝑥))𝑅)} = {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})
1615fveq2d 6415 . . . . . . . 8 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))
1712, 16eqeq12d 2814 . . . . . . 7 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → ((𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))
186, 17anbi12d 625 . . . . . 6 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})) ↔ ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))))
1918riotabidv 6841 . . . . 5 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))))
203, 19ifbieq2d 4302 . . . 4 (𝑥 = ⟨𝑋, 𝐹, 𝑌⟩ → if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
21 mapdh.i . . . 4 𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
22 mapdh.q . . . . . 6 𝑄 = (0g𝐶)
2322fvexi 6425 . . . . 5 𝑄 ∈ V
24 riotaex 6843 . . . . 5 (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))) ∈ V
2523, 24ifex 4325 . . . 4 if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))) ∈ V
2620, 21, 25fvmpt 6507 . . 3 (⟨𝑋, 𝐹, 𝑌⟩ ∈ V → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
271, 26mp1i 13 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
28 mapdh.y . . . . 5 (𝜑𝑌𝐸)
29 ot3rdg 7417 . . . . 5 (𝑌𝐸 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
3028, 29syl 17 . . . 4 (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
3130eqeq1d 2801 . . 3 (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0𝑌 = 0 ))
3230sneqd 4380 . . . . . . 7 (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
3332fveq2d 6415 . . . . . 6 (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
3433fveqeq2d 6419 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{})))
35 mapdh.x . . . . . . . . . . 11 (𝜑𝑋𝐴)
36 mapdh.f . . . . . . . . . . 11 (𝜑𝐹𝐵)
37 ot1stg 7415 . . . . . . . . . . 11 ((𝑋𝐴𝐹𝐵𝑌𝐸) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3835, 36, 28, 37syl3anc 1491 . . . . . . . . . 10 (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3938, 30oveq12d 6896 . . . . . . . . 9 (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 𝑌))
4039sneqd 4380 . . . . . . . 8 (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 𝑌)})
4140fveq2d 6415 . . . . . . 7 (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 𝑌)}))
4241fveq2d 6415 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 𝑌)})))
43 ot2ndg 7416 . . . . . . . . . 10 ((𝑋𝐴𝐹𝐵𝑌𝐸) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
4435, 36, 28, 43syl3anc 1491 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
4544oveq1d 6893 . . . . . . . 8 (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅) = (𝐹𝑅))
4645sneqd 4380 . . . . . . 7 (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)} = {(𝐹𝑅)})
4746fveq2d 6415 . . . . . 6 (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) = (𝐽‘{(𝐹𝑅)}))
4842, 47eqeq12d 2814 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) ↔ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))
4934, 48anbi12d 625 . . . 4 (𝜑 → (((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
5049riotabidv 6841 . . 3 (𝜑 → (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
5131, 50ifbieq2d 4302 . 2 (𝜑 → if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
5227, 51eqtrd 2833 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  ifcif 4277  {csn 4368  cotp 4376  cmpt 4922  cfv 6101  crio 6838  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  0gc0g 16415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-riota 6839  df-ov 6881  df-1st 7401  df-2nd 7402
This theorem is referenced by:  mapdhval0  37746  mapdhval2  37747  hdmap1valc  37824
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