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Theorem ovmpt2dxf 6228
 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 6127 . 2
3 ovmpt2dx.4 . . . 4
4 ovmpt2dxf.px . . . . 5
5 ovmpt2dx.5 . . . . . 6
6 ovmpt2dxf.py . . . . . . 7
7 eqid 2442 . . . . . . . . 9
87ovmpt4g 6225 . . . . . . . 8
98a1i 11 . . . . . . 7
106, 9alrimi 1783 . . . . . 6
115, 10spsbcd 3180 . . . . 5
124, 11alrimi 1783 . . . 4
133, 12spsbcd 3180 . . 3
145adantr 453 . . . . 5
15 simplr 733 . . . . . . . 8
163ad2antrr 708 . . . . . . . 8
1715, 16eqeltrd 2516 . . . . . . 7
185ad2antrr 708 . . . . . . . 8
19 simpr 449 . . . . . . . 8
20 ovmpt2dx.3 . . . . . . . . 9
2120adantr 453 . . . . . . . 8
2218, 19, 213eltr4d 2523 . . . . . . 7
23 ovmpt2dx.2 . . . . . . . . 9
2423anassrs 631 . . . . . . . 8
25 ovmpt2dx.6 . . . . . . . . . 10
26 elex 2970 . . . . . . . . . 10
2725, 26syl 16 . . . . . . . . 9
2827ad2antrr 708 . . . . . . . 8
2924, 28eqeltrd 2516 . . . . . . 7
30 biimt 327 . . . . . . 7
3117, 22, 29, 30syl3anc 1185 . . . . . 6
3215, 19oveq12d 6128 . . . . . . 7
3332, 24eqeq12d 2456 . . . . . 6
3431, 33bitr3d 248 . . . . 5
35 ovmpt2dxf.ay . . . . . . 7
3635nfeq2 2589 . . . . . 6
376, 36nfan 1848 . . . . 5
38 nfmpt22 6170 . . . . . . . 8
39 nfcv 2578 . . . . . . . 8
4035, 38, 39nfov 6133 . . . . . . 7
41 ovmpt2dxf.sy . . . . . . 7
4240, 41nfeq 2585 . . . . . 6
4342a1i 11 . . . . 5
4414, 34, 37, 43sbciedf 3202 . . . 4
45 nfcv 2578 . . . . . . 7
46 nfmpt21 6169 . . . . . . 7
47 ovmpt2dxf.bx . . . . . . 7
4845, 46, 47nfov 6133 . . . . . 6
49 ovmpt2dxf.sx . . . . . 6
5048, 49nfeq 2585 . . . . 5
5150a1i 11 . . . 4
523, 44, 4, 51sbciedf 3202 . . 3
5313, 52mpbid 203 . 2
542, 53eqtrd 2474 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937  wnf 1554   wceq 1653   wcel 1727  wnfc 2565  cvv 2962  wsbc 3167  (class class class)co 6110   cmpt2 6112 This theorem is referenced by:  ovmpt2dx  6229  mpt2xopoveq  6499  elovmpt2rab  28128  elovmpt2rab1  28129  ovmpt3rab1  28130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115
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