Theorem ovmpt4g 7566

Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 7005.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypothesis

Ref	Expression
ovmpt4g.3	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
ovmpt4g	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem ovmpt4g

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2847	. . 3 ⊢ (𝐶 ∈ 𝑉 → ∃𝑧 𝑧 = 𝐶)
2		moeq 3672	. . . . . . 7 ⊢ ∃*𝑧 𝑧 = 𝐶
3	2	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧 𝑧 = 𝐶)
4		ovmpt4g.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
5		df-mpo 7424	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	4, 5	eqtri 2788	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
7	3, 6	ovidi 7562	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8		eqeq2 2777	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
9	7, 8	mpbidi 244	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
10	9	exlimdv 1966	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
11	1, 10	syl5 35	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 ∈ 𝑉 → (𝑥𝐹𝑦) = 𝐶))
12	11	3impia 1135	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2567  (class class class)co 7419  {coprab 7420   ∈ cmpo 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424

This theorem is used by:  ovmpos  7567  ov2gf  7568  ovmpodxf  7569  ovmpodf  7575  ofmres  7987  fnmpoovd  8088  mapxpen  9138  pwfseqlem2  10661  pwfseqlem3  10662  fullfunc  17989  fthfunc  17990  prfcl  18283  curf1cl  18308  curfcl  18312  hofcl  18339  gsum2d2lem  20089  gsum2d2  20090  gsumcom2  20091  dprdval  20121  dprd2d2  20162  cnmpt21  23881  cnmpt2t  23883  cnmptcom  23888  cnmpt2k  23898  xkocnv  24024  suppovss  33099  fedgmullem1  34085  fedgmullem2  34086  fedgmul  34087  madjusmdetlem1  34283  madjusmdetlem3  34285  finxpreclem5  38100  sdclem2  38453  smflimlem1  47545  smflimlem2  47546  aovmpt4g  47998  ovmpordxf  49178  ovmpt4d  49702  iinfconstbas  49903  rescofuf  49930