Theorem ovmpt2x 5713

Description: The value of an operation class abstraction. Variant of ovmpt2ga 5714 which does not require D and x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Hypotheses

Ref	Expression
ovmpt2x.1	⊢ ((x = A ∧ y = B) → R = S)
ovmpt2x.2	⊢ (x = A → D = L)
ovmpt2x.3	⊢ F = (x ∈ C, y ∈ D ↦ R)

Assertion

Ref	Expression
ovmpt2x	⊢ ((A ∈ C ∧ B ∈ L ∧ S ∈ H) → (AFB) = S)

Distinct variable groups:   x,A,y   x,B,y   x,C,y   x,L,y   x,S,y

Allowed substitution hints:   D(x,y)   R(x,y)   F(x,y)   H(x,y)

Proof of Theorem ovmpt2x

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 952	. 2 ⊢ ((A ∈ C ∧ B ∈ L ∧ S ∈ H) → (A ∈ C ∧ B ∈ L))
2		ovmpt2x.1	. . . . . . 7 ⊢ ((x = A ∧ y = B) → R = S)
3	2	eqeq2d 2364	. . . . . 6 ⊢ ((x = A ∧ y = B) → (z = R ↔ z = S))
4	3	biimp3ar 1282	. . . . 5 ⊢ ((x = A ∧ y = B ∧ z = S) → z = R)
5	4	biantrud 493	. . . 4 ⊢ ((x = A ∧ y = B ∧ z = S) → ((x ∈ C ∧ y ∈ D) ↔ ((x ∈ C ∧ y ∈ D) ∧ z = R)))
6		simpl 443	. . . . . . 7 ⊢ ((x = A ∧ y = B) → x = A)
7	6	eleq1d 2419	. . . . . 6 ⊢ ((x = A ∧ y = B) → (x ∈ C ↔ A ∈ C))
8		ovmpt2x.2	. . . . . . . 8 ⊢ (x = A → D = L)
9	8	eleq2d 2420	. . . . . . 7 ⊢ (x = A → (y ∈ D ↔ y ∈ L))
10		eleq1 2413	. . . . . . 7 ⊢ (y = B → (y ∈ L ↔ B ∈ L))
11	9, 10	sylan9bb 680	. . . . . 6 ⊢ ((x = A ∧ y = B) → (y ∈ D ↔ B ∈ L))
12	7, 11	anbi12d 691	. . . . 5 ⊢ ((x = A ∧ y = B) → ((x ∈ C ∧ y ∈ D) ↔ (A ∈ C ∧ B ∈ L)))
13	12	3adant3 975	. . . 4 ⊢ ((x = A ∧ y = B ∧ z = S) → ((x ∈ C ∧ y ∈ D) ↔ (A ∈ C ∧ B ∈ L)))
14	5, 13	bitr3d 246	. . 3 ⊢ ((x = A ∧ y = B ∧ z = S) → (((x ∈ C ∧ y ∈ D) ∧ z = R) ↔ (A ∈ C ∧ B ∈ L)))
15		moeq 3013	. . . 4 ⊢ ∃*z z = R
16	15	moani 2256	. . 3 ⊢ ∃*z((x ∈ C ∧ y ∈ D) ∧ z = R)
17		ovmpt2x.3	. . . 4 ⊢ F = (x ∈ C, y ∈ D ↦ R)
18		df-mpt2 5655	. . . 4 ⊢ (x ∈ C, y ∈ D ↦ R) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ C ∧ y ∈ D) ∧ z = R)}
19	17, 18	eqtri 2373	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ C ∧ y ∈ D) ∧ z = R)}
20	14, 16, 19	ovigg 5597	. 2 ⊢ ((A ∈ C ∧ B ∈ L ∧ S ∈ H) → ((A ∈ C ∧ B ∈ L) → (AFB) = S))
21	1, 20	mpd 14	1 ⊢ ((A ∈ C ∧ B ∈ L ∧ S ∈ H) → (AFB) = S)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  (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: (None)