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Theorem 1st2val 6838
 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 4898 . . 3
2 fveq2 5879 . . . . . 6
3 df-ov 6311 . . . . . . 7
4 vex 3034 . . . . . . . 8
5 vex 3034 . . . . . . . 8
6 simpl 464 . . . . . . . . 9
7 mpt2v 6405 . . . . . . . . . 10
87eqcomi 2480 . . . . . . . . 9
96, 8, 4ovmpt2a 6446 . . . . . . . 8
104, 5, 9mp2an 686 . . . . . . 7
113, 10eqtr3i 2495 . . . . . 6
122, 11syl6eq 2521 . . . . 5
134, 5op1std 6822 . . . . 5
1412, 13eqtr4d 2508 . . . 4
1514exlimivv 1786 . . 3
161, 15sylbi 200 . 2
17 vex 3034 . . . . . . . . . 10
18 vex 3034 . . . . . . . . . 10
1917, 18pm3.2i 462 . . . . . . . . 9
20 ax6ev 1815 . . . . . . . . 9
2119, 202th 247 . . . . . . . 8
2221opabbii 4460 . . . . . . 7
23 df-xp 4845 . . . . . . 7
24 dmoprab 6396 . . . . . . 7
2522, 23, 243eqtr4ri 2504 . . . . . 6
2625eleq2i 2541 . . . . 5
27 ndmfv 5903 . . . . 5
2826, 27sylnbir 314 . . . 4
29 dmsnn0 5308 . . . . . . . 8
3029biimpri 211 . . . . . . 7
3130necon1bi 2671 . . . . . 6
3231unieqd 4200 . . . . 5
33 uni0 4217 . . . . 5
3432, 33syl6eq 2521 . . . 4
3528, 34eqtr4d 2508 . . 3
36 1stval 6814 . . 3
3735, 36syl6eqr 2523 . 2
3816, 37pm2.61i 169 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 376   wceq 1452  wex 1671   wcel 1904   wne 2641  cvv 3031  c0 3722  csn 3959  cop 3965  cuni 4190  copab 4453   cxp 4837   cdm 4839  cfv 5589  (class class class)co 6308  coprab 6309   cmpt2 6310  c1st 6810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812 This theorem is referenced by: (None)
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