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Theorem supiso 7239
 Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supiso.1
supiso.2
supisoex.3
supiso.4
Assertion
Ref Expression
supiso
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem supiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supiso.4 . . 3
2 supiso.1 . . . 4
3 isoso 5861 . . . 4
42, 3syl 15 . . 3
51, 4mpbid 201 . 2
6 isof1o 5838 . . . 4
7 f1of 5488 . . . 4
82, 6, 73syl 18 . . 3
9 supisoex.3 . . . 4
101, 9supcl 7225 . . 3
11 ffvelrn 5679 . . 3
128, 10, 11syl2anc 642 . 2
131, 9supub 7226 . . . . . 6
1413ralrimiv 2638 . . . . 5
151, 9suplub 7227 . . . . . . 7
1615exp3a 425 . . . . . 6
1716ralrimiv 2638 . . . . 5
18 supiso.2 . . . . . . 7
192, 18supisolem 7237 . . . . . 6
2010, 19mpdan 649 . . . . 5
2114, 17, 20mpbi2and 887 . . . 4
2221simpld 445 . . 3
2322r19.21bi 2654 . 2
2421simprd 449 . . . 4
2524r19.21bi 2654 . . 3
2625impr 602 . 2
275, 12, 23, 26eqsupd 7224 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  wral 2556  wrex 2557   wss 3165   class class class wbr 4039   wor 4329  cima 4708  wf 5267  wf1o 5270  cfv 5271   wiso 5272  csup 7209 This theorem is referenced by:  infmsup  9748 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-sup 7210
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