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Theorem ovmpt2dxf 6409
 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 6295 . 2
3 ovmpt2dx.4 . . . 4
4 ovmpt2dxf.px . . . . 5
5 ovmpt2dx.5 . . . . . 6
6 ovmpt2dxf.py . . . . . . 7
7 eqid 2402 . . . . . . . . 9
87ovmpt4g 6406 . . . . . . . 8
98a1i 11 . . . . . . 7
106, 9alrimi 1901 . . . . . 6
115, 10spsbcd 3291 . . . . 5
124, 11alrimi 1901 . . . 4
133, 12spsbcd 3291 . . 3
145adantr 463 . . . . 5
15 simplr 754 . . . . . . . 8
163ad2antrr 724 . . . . . . . 8
1715, 16eqeltrd 2490 . . . . . . 7
185ad2antrr 724 . . . . . . . 8
19 simpr 459 . . . . . . . 8
20 ovmpt2dx.3 . . . . . . . . 9
2120adantr 463 . . . . . . . 8
2218, 19, 213eltr4d 2505 . . . . . . 7
23 ovmpt2dx.2 . . . . . . . . 9
2423anassrs 646 . . . . . . . 8
25 ovmpt2dx.6 . . . . . . . . . 10
26 elex 3068 . . . . . . . . . 10
2725, 26syl 17 . . . . . . . . 9
2827ad2antrr 724 . . . . . . . 8
2924, 28eqeltrd 2490 . . . . . . 7
30 biimt 333 . . . . . . 7
3117, 22, 29, 30syl3anc 1230 . . . . . 6
3215, 19oveq12d 6296 . . . . . . 7
3332, 24eqeq12d 2424 . . . . . 6
3431, 33bitr3d 255 . . . . 5
35 ovmpt2dxf.ay . . . . . . 7
3635nfeq2 2581 . . . . . 6
376, 36nfan 1956 . . . . 5
38 nfmpt22 6346 . . . . . . . 8
39 nfcv 2564 . . . . . . . 8
4035, 38, 39nfov 6304 . . . . . . 7
41 ovmpt2dxf.sy . . . . . . 7
4240, 41nfeq 2575 . . . . . 6
4342a1i 11 . . . . 5
4414, 34, 37, 43sbciedf 3313 . . . 4
45 nfcv 2564 . . . . . . 7
46 nfmpt21 6345 . . . . . . 7
47 ovmpt2dxf.bx . . . . . . 7
4845, 46, 47nfov 6304 . . . . . 6
49 ovmpt2dxf.sx . . . . . 6
5048, 49nfeq 2575 . . . . 5
5150a1i 11 . . . 4
523, 44, 4, 51sbciedf 3313 . . 3
5313, 52mpbid 210 . 2
542, 53eqtrd 2443 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   w3a 974   wceq 1405  wnf 1637   wcel 1842  wnfc 2550  cvv 3059  wsbc 3277  (class class class)co 6278   cmpt2 6280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283 This theorem is referenced by:  ovmpt2dx  6410  elovmpt2rab  6502  elovmpt2rab1  6503  ovmpt3rab1  6515  mpt2xopoveq  6950  fvmpt2curryd  7003  mdetralt2  19403  mdetunilem2  19407  gsummatr01lem4  19452
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