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Theorem 2ndconst 8087
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
2ndconst (š“ āˆˆ š‘‰ ā†’ (2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“1-1-ontoā†’šµ)

Proof of Theorem 2ndconst
Dummy variables š‘„ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnzg 4779 . . 3 (š“ āˆˆ š‘‰ ā†’ {š“} ā‰  āˆ…)
2 fo2ndres 8002 . . 3 ({š“} ā‰  āˆ… ā†’ (2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“ontoā†’šµ)
31, 2syl 17 . 2 (š“ āˆˆ š‘‰ ā†’ (2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“ontoā†’šµ)
4 moeq 3704 . . . . . 6 āˆƒ*š‘„ š‘„ = āŸØš“, š‘¦āŸ©
54moani 2548 . . . . 5 āˆƒ*š‘„(š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)
6 vex 3479 . . . . . . . 8 š‘¦ āˆˆ V
76brresi 5991 . . . . . . 7 (š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦ ā†” (š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ š‘„2nd š‘¦))
8 fo2nd 7996 . . . . . . . . . . 11 2nd :Vā€“ontoā†’V
9 fofn 6808 . . . . . . . . . . 11 (2nd :Vā€“ontoā†’V ā†’ 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3479 . . . . . . . . . 10 š‘„ āˆˆ V
12 fnbrfvb 6945 . . . . . . . . . 10 ((2nd Fn V āˆ§ š‘„ āˆˆ V) ā†’ ((2nd ā€˜š‘„) = š‘¦ ā†” š‘„2nd š‘¦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((2nd ā€˜š‘„) = š‘¦ ā†” š‘„2nd š‘¦)
1413anbi2i 624 . . . . . . . 8 ((š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†” (š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ š‘„2nd š‘¦))
15 elxp7 8010 . . . . . . . . . . 11 (š‘„ āˆˆ ({š“} Ɨ šµ) ā†” (š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ {š“} āˆ§ (2nd ā€˜š‘„) āˆˆ šµ)))
16 eleq1 2822 . . . . . . . . . . . . . . 15 ((2nd ā€˜š‘„) = š‘¦ ā†’ ((2nd ā€˜š‘„) āˆˆ šµ ā†” š‘¦ āˆˆ šµ))
1716biimpac 480 . . . . . . . . . . . . . 14 (((2nd ā€˜š‘„) āˆˆ šµ āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ šµ)
1817adantll 713 . . . . . . . . . . . . 13 ((((1st ā€˜š‘„) āˆˆ {š“} āˆ§ (2nd ā€˜š‘„) āˆˆ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ šµ)
1918adantll 713 . . . . . . . . . . . 12 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ {š“} āˆ§ (2nd ā€˜š‘„) āˆˆ šµ)) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ šµ)
20 elsni 4646 . . . . . . . . . . . . . 14 ((1st ā€˜š‘„) āˆˆ {š“} ā†’ (1st ā€˜š‘„) = š“)
21 eqopi 8011 . . . . . . . . . . . . . . 15 ((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) = š“ āˆ§ (2nd ā€˜š‘„) = š‘¦)) ā†’ š‘„ = āŸØš“, š‘¦āŸ©)
2221anassrs 469 . . . . . . . . . . . . . 14 (((š‘„ āˆˆ (V Ɨ V) āˆ§ (1st ā€˜š‘„) = š“) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš“, š‘¦āŸ©)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((š‘„ āˆˆ (V Ɨ V) āˆ§ (1st ā€˜š‘„) āˆˆ {š“}) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš“, š‘¦āŸ©)
2423adantlrr 720 . . . . . . . . . . . 12 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ {š“} āˆ§ (2nd ā€˜š‘„) āˆˆ šµ)) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš“, š‘¦āŸ©)
2519, 24jca 513 . . . . . . . . . . 11 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ {š“} āˆ§ (2nd ā€˜š‘„) āˆˆ šµ)) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©))
2615, 25sylanb 582 . . . . . . . . . 10 ((š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†’ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©))
2726adantl 483 . . . . . . . . 9 ((š“ āˆˆ š‘‰ āˆ§ (š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦)) ā†’ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©))
28 simprr 772 . . . . . . . . . . 11 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ š‘„ = āŸØš“, š‘¦āŸ©)
29 snidg 4663 . . . . . . . . . . . . 13 (š“ āˆˆ š‘‰ ā†’ š“ āˆˆ {š“})
3029adantr 482 . . . . . . . . . . . 12 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ š“ āˆˆ {š“})
31 simprl 770 . . . . . . . . . . . 12 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ š‘¦ āˆˆ šµ)
3230, 31opelxpd 5716 . . . . . . . . . . 11 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ āŸØš“, š‘¦āŸ© āˆˆ ({š“} Ɨ šµ))
3328, 32eqeltrd 2834 . . . . . . . . . 10 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ š‘„ āˆˆ ({š“} Ɨ šµ))
34 fveq2 6892 . . . . . . . . . . . 12 (š‘„ = āŸØš“, š‘¦āŸ© ā†’ (2nd ā€˜š‘„) = (2nd ā€˜āŸØš“, š‘¦āŸ©))
35 op2ndg 7988 . . . . . . . . . . . . 13 ((š“ āˆˆ š‘‰ āˆ§ š‘¦ āˆˆ V) ā†’ (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
3635elvd 3482 . . . . . . . . . . . 12 (š“ āˆˆ š‘‰ ā†’ (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
3734, 36sylan9eqr 2795 . . . . . . . . . . 11 ((š“ āˆˆ š‘‰ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©) ā†’ (2nd ā€˜š‘„) = š‘¦)
3837adantrl 715 . . . . . . . . . 10 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ (2nd ā€˜š‘„) = š‘¦)
3933, 38jca 513 . . . . . . . . 9 ((š“ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)) ā†’ (š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦))
4027, 39impbida 800 . . . . . . . 8 (š“ āˆˆ š‘‰ ā†’ ((š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ (2nd ā€˜š‘„) = š‘¦) ā†” (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)))
4114, 40bitr3id 285 . . . . . . 7 (š“ āˆˆ š‘‰ ā†’ ((š‘„ āˆˆ ({š“} Ɨ šµ) āˆ§ š‘„2nd š‘¦) ā†” (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)))
427, 41bitrid 283 . . . . . 6 (š“ āˆˆ š‘‰ ā†’ (š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦ ā†” (š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)))
4342mobidv 2544 . . . . 5 (š“ āˆˆ š‘‰ ā†’ (āˆƒ*š‘„ š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦ ā†” āˆƒ*š‘„(š‘¦ āˆˆ šµ āˆ§ š‘„ = āŸØš“, š‘¦āŸ©)))
445, 43mpbiri 258 . . . 4 (š“ āˆˆ š‘‰ ā†’ āˆƒ*š‘„ š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦)
4544alrimiv 1931 . . 3 (š“ āˆˆ š‘‰ ā†’ āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦)
46 funcnv2 6617 . . 3 (Fun ā—”(2nd ā†¾ ({š“} Ɨ šµ)) ā†” āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ā†¾ ({š“} Ɨ šµ))š‘¦)
4745, 46sylibr 233 . 2 (š“ āˆˆ š‘‰ ā†’ Fun ā—”(2nd ā†¾ ({š“} Ɨ šµ)))
48 dff1o3 6840 . 2 ((2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“1-1-ontoā†’šµ ā†” ((2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“ontoā†’šµ āˆ§ Fun ā—”(2nd ā†¾ ({š“} Ɨ šµ))))
493, 47, 48sylanbrc 584 1 (š“ āˆˆ š‘‰ ā†’ (2nd ā†¾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)ā€“1-1-ontoā†’šµ)
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397  āˆ€wal 1540   = wceq 1542   āˆˆ wcel 2107  āˆƒ*wmo 2533   ā‰  wne 2941  Vcvv 3475  āˆ…c0 4323  {csn 4629  āŸØcop 4635   class class class wbr 5149   Ɨ cxp 5675  ā—”ccnv 5676   ā†¾ cres 5679  Fun wfun 6538   Fn wfn 6539  ā€“ontoā†’wfo 6542  ā€“1-1-ontoā†’wf1o 6543  ā€˜cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  curry1  8090  xpfiOLD  9318  fsum2dlem  15716  fprod2dlem  15924  gsum2dlem2  19839  ovoliunlem1  25019  gsummpt2d  32201  fv2ndcnv  34749
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