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Theorem 1st2val 6408
 Description: Value of an alternate definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
1st2val
Distinct variable group:   ,,
Allowed substitution hints:   (,,)

Proof of Theorem 1st2val
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4971 . . 3
2 fveq2 5763 . . . . . 6
3 df-ov 6120 . . . . . . 7
4 vex 2968 . . . . . . . 8
5 vex 2968 . . . . . . . 8
6 simpl 445 . . . . . . . . 9
7 mpt2v 6199 . . . . . . . . . 10
87eqcomi 2447 . . . . . . . . 9
96, 8, 4ovmpt2a 6240 . . . . . . . 8
104, 5, 9mp2an 655 . . . . . . 7
113, 10eqtr3i 2465 . . . . . 6
122, 11syl6eq 2491 . . . . 5
134, 5op1std 6393 . . . . 5
1412, 13eqtr4d 2478 . . . 4
1514exlimivv 1647 . . 3
161, 15sylbi 189 . 2
17 vex 2968 . . . . . . . . . 10
18 vex 2968 . . . . . . . . . 10
1917, 18pm3.2i 443 . . . . . . . . 9
20 a9ev 1671 . . . . . . . . 9
2119, 202th 232 . . . . . . . 8
2221opabbii 4303 . . . . . . 7
23 df-xp 4919 . . . . . . 7
24 dmoprab 6190 . . . . . . 7
2522, 23, 243eqtr4ri 2474 . . . . . 6
2625eleq2i 2507 . . . . 5
27 ndmfv 5786 . . . . 5
2826, 27sylnbir 300 . . . 4
29 dmsnn0 5370 . . . . . . . 8
3029biimpri 199 . . . . . . 7
3130necon1bi 2654 . . . . . 6
3231unieqd 4055 . . . . 5
33 uni0 4071 . . . . 5
3432, 33syl6eq 2491 . . . 4
3528, 34eqtr4d 2478 . . 3
36 1stval 6387 . . 3
3735, 36syl6eqr 2493 . 2
3816, 37pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 360  wex 1551   wceq 1654   wcel 1728   wne 2606  cvv 2965  c0 3616  csn 3843  cop 3846  cuni 4044  copab 4296   cxp 4911   cdm 4913  cfv 5489  (class class class)co 6117  coprab 6118   cmpt2 6119  c1st 6383 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 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736 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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385
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