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Theorem ovmpt2x 5712
 Description: The value of an operation class abstraction. Variant of ovmpt2ga 5713 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 = AD = 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.
StepHypRef 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)
32eqeq2d 2364 . . . . . 6 ((x = A y = B) → (z = Rz = S))
43biimp3ar 1282 . . . . 5 ((x = A y = B z = S) → z = R)
54biantrud 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)
76eleq1d 2419 . . . . . 6 ((x = A y = B) → (x CA C))
8 ovmpt2x.2 . . . . . . . 8 (x = AD = L)
98eleq2d 2420 . . . . . . 7 (x = A → (y Dy L))
10 eleq1 2413 . . . . . . 7 (y = B → (y LB L))
119, 10sylan9bb 680 . . . . . 6 ((x = A y = B) → (y DB L))
127, 11anbi12d 691 . . . . 5 ((x = A y = B) → ((x C y D) ↔ (A C B L)))
13123adant3 975 . . . 4 ((x = A y = B z = S) → ((x C y D) ↔ (A C B L)))
145, 13bitr3d 246 . . 3 ((x = A y = B z = S) → (((x C y D) z = R) ↔ (A C B L)))
15 moeq 3012 . . . 4 ∃*z z = R
1615moani 2256 . . 3 ∃*z((x C y D) z = R)
17 ovmpt2x.3 . . . 4 F = (x C, y D R)
18 df-mpt2 5654 . . . 4 (x C, y D R) = {x, y, z ((x C y D) z = R)}
1917, 18eqtri 2373 . . 3 F = {x, y, z ((x C y D) z = R)}
2014, 16, 19ovigg 5596 . 2 ((A C B L S H) → ((A C B L) → (AFB) = S))
211, 20mpd 14 1 ((A C B L S H) → (AFB) = S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  (class class class)co 5525  {coprab 5527   ↦ cmpt2 5653 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528  df-mpt2 5654 This theorem is referenced by: (None)
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