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Theorem ovmpt2dxf 6057
 Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovmpt2dx.1
ovmpt2dx.2
ovmpt2dx.3
ovmpt2dx.4
ovmpt2dx.5
ovmpt2dx.6
ovmpt2dxf.px
ovmpt2dxf.py
ovmpt2dxf.ay
ovmpt2dxf.bx
ovmpt2dxf.sx
ovmpt2dxf.sy
Assertion
Ref Expression
ovmpt2dxf
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   (,)   ()   ()   (,)   (,)   (,)   (,)   (,)   (,)   (,)

Proof of Theorem ovmpt2dxf
StepHypRef Expression
1 ovmpt2dx.1 . . 3
21oveqd 5959 . 2
3 ovmpt2dx.4 . . . 4
4 ovmpt2dxf.px . . . . 5
5 ovmpt2dx.5 . . . . . 6
6 ovmpt2dxf.py . . . . . . 7
7 eqid 2358 . . . . . . . . 9
87ovmpt4g 6054 . . . . . . . 8
98a1i 10 . . . . . . 7
106, 9alrimi 1766 . . . . . 6
11 spsbc 3079 . . . . . 6
125, 10, 11sylc 56 . . . . 5
134, 12alrimi 1766 . . . 4
14 spsbc 3079 . . . 4
153, 13, 14sylc 56 . . 3
165adantr 451 . . . . 5
17 simplr 731 . . . . . . . 8
183ad2antrr 706 . . . . . . . 8
1917, 18eqeltrd 2432 . . . . . . 7
205ad2antrr 706 . . . . . . . 8
21 simpr 447 . . . . . . . . 9
22 ovmpt2dx.3 . . . . . . . . . 10
2322adantr 451 . . . . . . . . 9
2421, 23eleq12d 2426 . . . . . . . 8
2520, 24mpbird 223 . . . . . . 7
26 ovmpt2dx.2 . . . . . . . . 9
2726anassrs 629 . . . . . . . 8
28 ovmpt2dx.6 . . . . . . . . . 10
29 elex 2872 . . . . . . . . . 10
3028, 29syl 15 . . . . . . . . 9
3130ad2antrr 706 . . . . . . . 8
3227, 31eqeltrd 2432 . . . . . . 7
33 biimt 325 . . . . . . 7
3419, 25, 32, 33syl3anc 1182 . . . . . 6
3517, 21oveq12d 5960 . . . . . . 7
3635, 27eqeq12d 2372 . . . . . 6
3734, 36bitr3d 246 . . . . 5
38 ovmpt2dxf.ay . . . . . . 7
3938nfeq2 2505 . . . . . 6
406, 39nfan 1829 . . . . 5
41 nfmpt22 5999 . . . . . . . 8
42 nfcv 2494 . . . . . . . 8
4338, 41, 42nfov 5965 . . . . . . 7
44 ovmpt2dxf.sy . . . . . . 7
4543, 44nfeq 2501 . . . . . 6
4645a1i 10 . . . . 5
4716, 37, 40, 46sbciedf 3102 . . . 4
48 nfcv 2494 . . . . . . 7
49 nfmpt21 5998 . . . . . . 7
50 ovmpt2dxf.bx . . . . . . 7
5148, 49, 50nfov 5965 . . . . . 6
52 ovmpt2dxf.sx . . . . . 6
5351, 52nfeq 2501 . . . . 5
5453a1i 10 . . . 4
553, 47, 4, 54sbciedf 3102 . . 3
5615, 55mpbid 201 . 2
572, 56eqtrd 2390 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934  wal 1540  wnf 1544   wceq 1642   wcel 1710  wnfc 2481  cvv 2864  wsbc 3067  (class class class)co 5942   cmpt2 5944 This theorem is referenced by:  ovmpt2dx  6058  fmuldfeqlem1  27035  mpt2xopoveq  27439 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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947
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