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Theorem spwval2 14648
 Description: Value of supremum under a weak ordering. Read as "the -supremum of ." is the field of a relation by relfld 5387. Unlike df-sup 7438 for strong orderings, the supremum exists iff belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
Hypothesis
Ref Expression
spwval2.1
Assertion
Ref Expression
spwval2
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem spwval2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2
2 unieq 4016 . . . . . 6
32unieqd 4018 . . . . 5
4 spwval2.1 . . . . 5
53, 4syl6eqr 2485 . . . 4
6 breq 4206 . . . . . 6
76ralbidv 2717 . . . . 5
8 breq 4206 . . . . . . . 8
98ralbidv 2717 . . . . . . 7
10 breq 4206 . . . . . . 7
119, 10imbi12d 312 . . . . . 6
125, 11raleqbidv 2908 . . . . 5
137, 12anbi12d 692 . . . 4
145, 13riotaeqbidv 6544 . . 3
15 raleq 2896 . . . . 5
16 raleq 2896 . . . . . . 7
1716imbi1d 309 . . . . . 6
1817ralbidv 2717 . . . . 5
1915, 18anbi12d 692 . . . 4
2019riotabidv 6543 . . 3
21 df-spw 14623 . . 3
22 riotaex 6545 . . 3
2314, 20, 21, 22ovmpt2 6201 . 2
241, 23sylan2 461 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948  cuni 4007   class class class wbr 4204  (class class class)co 6073  crio 6534  cps 14616   cspw 14618 This theorem is referenced by:  spwval  14649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-spw 14623
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