Theorem ovmpt4g 7536

Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 6979.) (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 2810	. . 3 ⊢ (𝐶 ∈ 𝑉 → ∃𝑧 𝑧 = 𝐶)
2		moeq 3678	. . . . . . 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 7392	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	4, 5	eqtri 2752	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
7	3, 6	ovidi 7532	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8		eqeq2 2741	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
9	7, 8	mpbidi 241	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
10	9	exlimdv 1933	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
11	1, 10	syl5 34	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 ∈ 𝑉 → (𝑥𝐹𝑦) = 𝐶))
12	11	3impia 1117	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  (class class class)co 7387  {coprab 7388   ∈ cmpo 7389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392

This theorem is referenced by:  ovmpos  7537  ov2gf  7538  ovmpodxf  7539  ovmpodf  7545  ofmres  7963  fnmpoovd  8066  mapxpen  9107  pwfseqlem2  10612  pwfseqlem3  10613  fullfunc  17870  fthfunc  17871  prfcl  18164  curf1cl  18189  curfcl  18193  hofcl  18220  gsum2d2lem  19903  gsum2d2  19904  gsumcom2  19905  dprdval  19935  dprd2d2  19976  cnmpt21  23558  cnmpt2t  23560  cnmptcom  23565  cnmpt2k  23575  xkocnv  23701  suppovss  32604  fedgmullem1  33625  fedgmullem2  33626  fedgmul  33627  madjusmdetlem1  33817  madjusmdetlem3  33819  finxpreclem5  37383  sdclem2  37736  smflimlem1  46769  smflimlem2  46770  aovmpt4g  47202  ovmpordxf  48327  ovmpt4d  48853  iinfconstbas  49055  rescofuf  49082