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Theorem ovmpt2dxf 6428
 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:   ,   ,   ,

Proof of Theorem ovmpt2dxf
StepHypRef Expression
1 ovmpt2dx.1 . . 3
21oveqd 6313 . 2
3 ovmpt2dx.4 . . . 4
4 ovmpt2dxf.px . . . . 5
5 ovmpt2dx.5 . . . . . 6
6 ovmpt2dxf.py . . . . . . 7
7 eqid 2457 . . . . . . . . 9
87ovmpt4g 6425 . . . . . . . 8
98a1i 11 . . . . . . 7
106, 9alrimi 1877 . . . . . 6
115, 10spsbcd 3341 . . . . 5
124, 11alrimi 1877 . . . 4
133, 12spsbcd 3341 . . 3
145adantr 465 . . . . 5
15 simplr 755 . . . . . . . 8
163ad2antrr 725 . . . . . . . 8
1715, 16eqeltrd 2545 . . . . . . 7
185ad2antrr 725 . . . . . . . 8
19 simpr 461 . . . . . . . 8
20 ovmpt2dx.3 . . . . . . . . 9
2120adantr 465 . . . . . . . 8
2218, 19, 213eltr4d 2560 . . . . . . 7
23 ovmpt2dx.2 . . . . . . . . 9
2423anassrs 648 . . . . . . . 8
25 ovmpt2dx.6 . . . . . . . . . 10
26 elex 3118 . . . . . . . . . 10
2725, 26syl 16 . . . . . . . . 9
2827ad2antrr 725 . . . . . . . 8
2924, 28eqeltrd 2545 . . . . . . 7
30 biimt 335 . . . . . . 7
3117, 22, 29, 30syl3anc 1228 . . . . . 6
3215, 19oveq12d 6314 . . . . . . 7
3332, 24eqeq12d 2479 . . . . . 6
3431, 33bitr3d 255 . . . . 5
35 ovmpt2dxf.ay . . . . . . 7
3635nfeq2 2636 . . . . . 6
376, 36nfan 1928 . . . . 5
38 nfmpt22 6365 . . . . . . . 8
39 nfcv 2619 . . . . . . . 8
4035, 38, 39nfov 6322 . . . . . . 7
41 ovmpt2dxf.sy . . . . . . 7
4240, 41nfeq 2630 . . . . . 6
4342a1i 11 . . . . 5
4414, 34, 37, 43sbciedf 3363 . . . 4
45 nfcv 2619 . . . . . . 7
46 nfmpt21 6364 . . . . . . 7
47 ovmpt2dxf.bx . . . . . . 7
4845, 46, 47nfov 6322 . . . . . 6
49 ovmpt2dxf.sx . . . . . 6
5048, 49nfeq 2630 . . . . 5
5150a1i 11 . . . 4
523, 44, 4, 51sbciedf 3363 . . 3
5313, 52mpbid 210 . 2
542, 53eqtrd 2498 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  F/wnf 1616  e.wcel 1818  F/_wnfc 2605   cvv 3109  [.wsbc 3327  (class class class)co 6296  e.cmpt2 6298 This theorem is referenced by:  ovmpt2dx  6429  elovmpt2rab  6521  elovmpt2rab1  6522  ovmpt3rab1  6534  mpt2xopoveq  6966  fvmpt2curryd  7019  mdetralt2  19111  mdetunilem2  19115  gsummatr01lem4  19160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301
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