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Theorem ov2gf 5711
 Description: The value of an operation class abstraction. A version of ovmpt2g 5715 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ov2gf.a xA
ov2gf.c yA
ov2gf.d yB
ov2gf.1 xG
ov2gf.2 yS
ov2gf.3 (x = AR = G)
ov2gf.4 (y = BG = S)
ov2gf.5 F = (x C, y D R)
Assertion
Ref Expression
ov2gf ((A C B D S H) → (AFB) = S)
Distinct variable groups:   x,y,C   x,D,y
Allowed substitution hints:   A(x,y)   B(x,y)   R(x,y)   S(x,y)   F(x,y)   G(x,y)   H(x,y)

Proof of Theorem ov2gf
StepHypRef Expression
1 elex 2867 . . 3 (S HS V)
2 ov2gf.a . . . 4 xA
3 ov2gf.c . . . 4 yA
4 ov2gf.d . . . 4 yB
5 ov2gf.1 . . . . . 6 xG
65nfel1 2499 . . . . 5 x G V
7 ov2gf.5 . . . . . . . 8 F = (x C, y D R)
8 nfmpt21 5673 . . . . . . . 8 x(x C, y D R)
97, 8nfcxfr 2486 . . . . . . 7 xF
10 nfcv 2489 . . . . . . 7 xy
112, 9, 10nfov 5545 . . . . . 6 x(AFy)
1211, 5nfeq 2496 . . . . 5 x(AFy) = G
136, 12nfim 1813 . . . 4 x(G V → (AFy) = G)
14 ov2gf.2 . . . . . 6 yS
1514nfel1 2499 . . . . 5 y S V
16 nfmpt22 5674 . . . . . . . 8 y(x C, y D R)
177, 16nfcxfr 2486 . . . . . . 7 yF
183, 17, 4nfov 5545 . . . . . 6 y(AFB)
1918, 14nfeq 2496 . . . . 5 y(AFB) = S
2015, 19nfim 1813 . . . 4 y(S V → (AFB) = S)
21 ov2gf.3 . . . . . 6 (x = AR = G)
2221eleq1d 2419 . . . . 5 (x = A → (R V ↔ G V))
23 oveq1 5530 . . . . . 6 (x = A → (xFy) = (AFy))
2423, 21eqeq12d 2367 . . . . 5 (x = A → ((xFy) = R ↔ (AFy) = G))
2522, 24imbi12d 311 . . . 4 (x = A → ((R V → (xFy) = R) ↔ (G V → (AFy) = G)))
26 ov2gf.4 . . . . . 6 (y = BG = S)
2726eleq1d 2419 . . . . 5 (y = B → (G V ↔ S V))
28 oveq2 5531 . . . . . 6 (y = B → (AFy) = (AFB))
2928, 26eqeq12d 2367 . . . . 5 (y = B → ((AFy) = G ↔ (AFB) = S))
3027, 29imbi12d 311 . . . 4 (y = B → ((G V → (AFy) = G) ↔ (S V → (AFB) = S)))
317ovmpt4g 5710 . . . . 5 ((x C y D R V) → (xFy) = R)
32313expia 1153 . . . 4 ((x C y D) → (R V → (xFy) = R))
332, 3, 4, 13, 20, 25, 30, 32vtocl2gaf 2921 . . 3 ((A C B D) → (S V → (AFB) = S))
341, 33syl5 28 . 2 ((A C B D) → (S H → (AFB) = S))
35343impia 1148 1 ((A C B D S H) → (AFB) = S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  (class class class)co 5525   ↦ 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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