Theorem ovmpt4g 7539

Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 6982.) (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 2811	. . 3 ⊢ (𝐶 ∈ 𝑉 → ∃𝑧 𝑧 = 𝐶)
2		moeq 3681	. . . . . . 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 7395	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
6	4, 5	eqtri 2753	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
7	3, 6	ovidi 7535	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8		eqeq2 2742	. . . . 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 2532  (class class class)co 7390  {coprab 7391   ∈ cmpo 7392

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  ovmpos  7540  ov2gf  7541  ovmpodxf  7542  ovmpodf  7548  ofmres  7966  fnmpoovd  8069  mapxpen  9113  pwfseqlem2  10619  pwfseqlem3  10620  fullfunc  17877  fthfunc  17878  prfcl  18171  curf1cl  18196  curfcl  18200  hofcl  18227  gsum2d2lem  19910  gsum2d2  19911  gsumcom2  19912  dprdval  19942  dprd2d2  19983  cnmpt21  23565  cnmpt2t  23567  cnmptcom  23572  cnmpt2k  23582  xkocnv  23708  suppovss  32611  fedgmullem1  33632  fedgmullem2  33633  fedgmul  33634  madjusmdetlem1  33824  madjusmdetlem3  33826  finxpreclem5  37390  sdclem2  37743  smflimlem1  46776  smflimlem2  46777  aovmpt4g  47206  ovmpordxf  48331  ovmpt4d  48857  iinfconstbas  49059  rescofuf  49086