Theorem ovmpt4g 7559

Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 7002.) (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 2817	. . 3 ⊢ (𝐶 ∈ 𝑉 → ∃𝑧 𝑧 = 𝐶)
2		moeq 3695	. . . . . . 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 7415	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	4, 5	eqtri 2759	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
7	3, 6	ovidi 7555	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8		eqeq2 2748	. . . . 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 2538  (class class class)co 7410  {coprab 7411   ∈ cmpo 7412

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  ovmpos  7560  ov2gf  7561  ovmpodxf  7562  ovmpodf  7568  ofmres  7988  fnmpoovd  8091  mapxpen  9162  pwfseqlem2  10678  pwfseqlem3  10679  fullfunc  17926  fthfunc  17927  prfcl  18220  curf1cl  18245  curfcl  18249  hofcl  18276  gsum2d2lem  19959  gsum2d2  19960  gsumcom2  19961  dprdval  19991  dprd2d2  20032  cnmpt21  23614  cnmpt2t  23616  cnmptcom  23621  cnmpt2k  23631  xkocnv  23757  suppovss  32663  fedgmullem1  33674  fedgmullem2  33675  fedgmul  33676  madjusmdetlem1  33863  madjusmdetlem3  33865  finxpreclem5  37418  sdclem2  37771  smflimlem1  46767  smflimlem2  46768  aovmpt4g  47197  ovmpordxf  48281  ovmpt4d  48808  iinfconstbas  49000  rescofuf  49023