MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stconst Structured version   Visualization version   GIF version

Theorem 1stconst 8033
Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
1stconst (šµ āˆˆ š‘‰ ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“1-1-ontoā†’š“)

Proof of Theorem 1stconst
Dummy variables š‘„ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnzg 4736 . . 3 (šµ āˆˆ š‘‰ ā†’ {šµ} ā‰  āˆ…)
2 fo1stres 7948 . . 3 ({šµ} ā‰  āˆ… ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“ontoā†’š“)
31, 2syl 17 . 2 (šµ āˆˆ š‘‰ ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“ontoā†’š“)
4 moeq 3666 . . . . . 6 āˆƒ*š‘„ š‘„ = āŸØš‘¦, šµāŸ©
54moani 2548 . . . . 5 āˆƒ*š‘„(š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)
6 vex 3448 . . . . . . . 8 š‘¦ āˆˆ V
76brresi 5947 . . . . . . 7 (š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦ ā†” (š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ š‘„1st š‘¦))
8 fo1st 7942 . . . . . . . . . . 11 1st :Vā€“ontoā†’V
9 fofn 6759 . . . . . . . . . . 11 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3448 . . . . . . . . . 10 š‘„ āˆˆ V
12 fnbrfvb 6896 . . . . . . . . . 10 ((1st Fn V āˆ§ š‘„ āˆˆ V) ā†’ ((1st ā€˜š‘„) = š‘¦ ā†” š‘„1st š‘¦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((1st ā€˜š‘„) = š‘¦ ā†” š‘„1st š‘¦)
1413anbi2i 624 . . . . . . . 8 ((š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†” (š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ š‘„1st š‘¦))
15 elxp7 7957 . . . . . . . . . . 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 ((1st ā€˜š‘„) = š‘¦ ā†’ ((1st ā€˜š‘„) āˆˆ š“ ā†” š‘¦ āˆˆ š“))
1716biimpac 480 . . . . . . . . . . . . . 14 (((1st ā€˜š‘„) āˆˆ š“ āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ š“)
1817adantlr 714 . . . . . . . . . . . . 13 ((((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ š“)
1918adantll 713 . . . . . . . . . . . 12 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ})) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘¦ āˆˆ š“)
20 elsni 4604 . . . . . . . . . . . . . 14 ((2nd ā€˜š‘„) āˆˆ {šµ} ā†’ (2nd ā€˜š‘„) = šµ)
21 eqopi 7958 . . . . . . . . . . . . . . 15 ((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) = š‘¦ āˆ§ (2nd ā€˜š‘„) = šµ)) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2221anass1rs 654 . . . . . . . . . . . . . 14 (((š‘„ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘„) = šµ) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((š‘„ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2423adantlrl 719 . . . . . . . . . . . 12 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ})) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2519, 24jca 513 . . . . . . . . . . 11 (((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ})) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
2615, 25sylanb 582 . . . . . . . . . 10 ((š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
2726adantl 483 . . . . . . . . 9 ((šµ āˆˆ š‘‰ āˆ§ (š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦)) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
28 simprr 772 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
29 simprl 770 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘¦ āˆˆ š“)
30 snidg 4621 . . . . . . . . . . . . 13 (šµ āˆˆ š‘‰ ā†’ šµ āˆˆ {šµ})
3130adantr 482 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ šµ āˆˆ {šµ})
3229, 31opelxpd 5672 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ āŸØš‘¦, šµāŸ© āˆˆ (š“ Ɨ {šµ}))
3328, 32eqeltrd 2834 . . . . . . . . . 10 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘„ āˆˆ (š“ Ɨ {šµ}))
3428fveq2d 6847 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜š‘„) = (1st ā€˜āŸØš‘¦, šµāŸ©))
35 simpl 484 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ šµ āˆˆ š‘‰)
36 op1stg 7934 . . . . . . . . . . . 12 ((š‘¦ āˆˆ š“ āˆ§ šµ āˆˆ š‘‰) ā†’ (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3729, 35, 36syl2anc 585 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3834, 37eqtrd 2773 . . . . . . . . . 10 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜š‘„) = š‘¦)
3933, 38jca 513 . . . . . . . . 9 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦))
4027, 39impbida 800 . . . . . . . 8 (šµ āˆˆ š‘‰ ā†’ ((š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ (1st ā€˜š‘„) = š‘¦) ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
4114, 40bitr3id 285 . . . . . . 7 (šµ āˆˆ š‘‰ ā†’ ((š‘„ āˆˆ (š“ Ɨ {šµ}) āˆ§ š‘„1st š‘¦) ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
427, 41bitrid 283 . . . . . 6 (šµ āˆˆ š‘‰ ā†’ (š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦ ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
4342mobidv 2544 . . . . 5 (šµ āˆˆ š‘‰ ā†’ (āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦ ā†” āˆƒ*š‘„(š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
445, 43mpbiri 258 . . . 4 (šµ āˆˆ š‘‰ ā†’ āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
4544alrimiv 1931 . . 3 (šµ āˆˆ š‘‰ ā†’ āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
46 funcnv2 6570 . . 3 (Fun ā—”(1st ā†¾ (š“ Ɨ {šµ})) ā†” āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
4745, 46sylibr 233 . 2 (šµ āˆˆ š‘‰ ā†’ Fun ā—”(1st ā†¾ (š“ Ɨ {šµ})))
48 dff1o3 6791 . 2 ((1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“1-1-ontoā†’š“ ā†” ((1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“ontoā†’š“ āˆ§ Fun ā—”(1st ā†¾ (š“ Ɨ {šµ}))))
493, 47, 48sylanbrc 584 1 (šµ āˆˆ š‘‰ ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“1-1-ontoā†’š“)
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 2940  Vcvv 3444  āˆ…c0 4283  {csn 4587  āŸØcop 4593   class class class wbr 5106   Ɨ cxp 5632  ā—”ccnv 5633   ā†¾ cres 5636  Fun wfun 6491   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€“1-1-ontoā†’wf1o 6496  ā€˜cfv 6497  1st c1st 7920  2nd c2nd 7921
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 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  curry2  8040  domss2  9083  fv1stcnv  34407
  Copyright terms: Public domain W3C validator