Theorem ovmpt4g 5711

Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5705.) (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 2870	. . 3 ⊢ (C ∈ V → ∃z z = C)
2		moeq 3013	. . . . . . 7 ⊢ ∃*z z = C
3	2	a1i 10	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ B) → ∃*z z = C)
4		ovmpt4g.3	. . . . . . 7 ⊢ F = (x ∈ A, y ∈ B ↦ C)
5		df-mpt2 5655	. . . . . . 7 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
6	4, 5	eqtri 2373	. . . . . 6 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ z = C)}
7	3, 6	ovidi 5595	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) → (z = C → (xFy) = z))
8		eqeq2 2362	. . . . 5 ⊢ (z = C → ((xFy) = z ↔ (xFy) = C))
9	7, 8	mpbidi 207	. . . 4 ⊢ ((x ∈ A ∧ y ∈ B) → (z = C → (xFy) = C))
10	9	exlimdv 1636	. . 3 ⊢ ((x ∈ A ∧ y ∈ B) → (∃z z = C → (xFy) = C))
11	1, 10	syl5 28	. 2 ⊢ ((x ∈ A ∧ y ∈ B) → (C ∈ V → (xFy) = C))
12	11	3impia 1148	1 ⊢ ((x ∈ A ∧ y ∈ B ∧ C ∈ V) → (xFy) = C)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  (class class class)co 5526  {coprab 5528   ↦ cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796  df-ov 5527  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  ov2gf  5712