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Theorem 1stconst 6224
Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
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 snmg 3712 . . 3 (šµ āˆˆ š‘‰ ā†’ āˆƒš‘„ š‘„ āˆˆ {šµ})
2 fo1stresm 6164 . . 3 (āˆƒš‘„ š‘„ āˆˆ {šµ} ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“ontoā†’š“)
31, 2syl 14 . 2 (šµ āˆˆ š‘‰ ā†’ (1st ā†¾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})ā€“ontoā†’š“)
4 moeq 2914 . . . . . 6 āˆƒ*š‘„ š‘„ = āŸØš‘¦, šµāŸ©
54moani 2096 . . . . 5 āˆƒ*š‘„(š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)
6 vex 2742 . . . . . . . 8 š‘¦ āˆˆ V
76brres 4915 . . . . . . 7 (š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦ ā†” (š‘„1st š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})))
8 fo1st 6160 . . . . . . . . . . 11 1st :Vā€“ontoā†’V
9 fofn 5442 . . . . . . . . . . 11 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 2742 . . . . . . . . . 10 š‘„ āˆˆ V
12 fnbrfvb 5558 . . . . . . . . . 10 ((1st Fn V āˆ§ š‘„ āˆˆ V) ā†’ ((1st ā€˜š‘„) = š‘¦ ā†” š‘„1st š‘¦))
1310, 11, 12mp2an 426 . . . . . . . . 9 ((1st ā€˜š‘„) = š‘¦ ā†” š‘„1st š‘¦)
1413anbi1i 458 . . . . . . . 8 (((1st ā€˜š‘„) = š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})) ā†” (š‘„1st š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})))
15 elxp7 6173 . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . 15 ((1st ā€˜š‘„) = š‘¦ ā†’ ((1st ā€˜š‘„) āˆˆ š“ ā†” š‘¦ āˆˆ š“))
1716biimpa 296 . . . . . . . . . . . . . 14 (((1st ā€˜š‘„) = š‘¦ āˆ§ (1st ā€˜š‘„) āˆˆ š“) ā†’ š‘¦ āˆˆ š“)
1817adantrr 479 . . . . . . . . . . . . 13 (((1st ā€˜š‘„) = š‘¦ āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ})) ā†’ š‘¦ āˆˆ š“)
1918adantrl 478 . . . . . . . . . . . 12 (((1st ā€˜š‘„) = š‘¦ āˆ§ (š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ}))) ā†’ š‘¦ āˆˆ š“)
20 elsni 3612 . . . . . . . . . . . . . 14 ((2nd ā€˜š‘„) āˆˆ {šµ} ā†’ (2nd ā€˜š‘„) = šµ)
21 eqopi 6175 . . . . . . . . . . . . . . 15 ((š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) = š‘¦ āˆ§ (2nd ā€˜š‘„) = šµ)) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2221an12s 565 . . . . . . . . . . . . . 14 (((1st ā€˜š‘„) = š‘¦ āˆ§ (š‘„ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘„) = šµ)) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2320, 22sylanr2 405 . . . . . . . . . . . . 13 (((1st ā€˜š‘„) = š‘¦ āˆ§ (š‘„ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ})) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2423adantrrl 486 . . . . . . . . . . . 12 (((1st ā€˜š‘„) = š‘¦ āˆ§ (š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ}))) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2519, 24jca 306 . . . . . . . . . . 11 (((1st ā€˜š‘„) = š‘¦ āˆ§ (š‘„ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘„) āˆˆ š“ āˆ§ (2nd ā€˜š‘„) āˆˆ {šµ}))) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
2615, 25sylan2b 287 . . . . . . . . . 10 (((1st ā€˜š‘„) = š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
2726adantl 277 . . . . . . . . 9 ((šµ āˆˆ š‘‰ āˆ§ ((1st ā€˜š‘„) = š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ}))) ā†’ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©))
28 simprr 531 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘„ = āŸØš‘¦, šµāŸ©)
2928fveq2d 5521 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜š‘„) = (1st ā€˜āŸØš‘¦, šµāŸ©))
30 simprl 529 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘¦ āˆˆ š“)
31 simpl 109 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ šµ āˆˆ š‘‰)
32 op1stg 6153 . . . . . . . . . . . 12 ((š‘¦ āˆˆ š“ āˆ§ šµ āˆˆ š‘‰) ā†’ (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3330, 31, 32syl2anc 411 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3429, 33eqtrd 2210 . . . . . . . . . 10 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ (1st ā€˜š‘„) = š‘¦)
35 snidg 3623 . . . . . . . . . . . . 13 (šµ āˆˆ š‘‰ ā†’ šµ āˆˆ {šµ})
3635adantr 276 . . . . . . . . . . . 12 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ šµ āˆˆ {šµ})
37 opelxpi 4660 . . . . . . . . . . . 12 ((š‘¦ āˆˆ š“ āˆ§ šµ āˆˆ {šµ}) ā†’ āŸØš‘¦, šµāŸ© āˆˆ (š“ Ɨ {šµ}))
3830, 36, 37syl2anc 411 . . . . . . . . . . 11 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ āŸØš‘¦, šµāŸ© āˆˆ (š“ Ɨ {šµ}))
3928, 38eqeltrd 2254 . . . . . . . . . 10 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ š‘„ āˆˆ (š“ Ɨ {šµ}))
4034, 39jca 306 . . . . . . . . 9 ((šµ āˆˆ š‘‰ āˆ§ (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)) ā†’ ((1st ā€˜š‘„) = š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})))
4127, 40impbida 596 . . . . . . . 8 (šµ āˆˆ š‘‰ ā†’ (((1st ā€˜š‘„) = š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})) ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
4214, 41bitr3id 194 . . . . . . 7 (šµ āˆˆ š‘‰ ā†’ ((š‘„1st š‘¦ āˆ§ š‘„ āˆˆ (š“ Ɨ {šµ})) ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
437, 42bitrid 192 . . . . . 6 (šµ āˆˆ š‘‰ ā†’ (š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦ ā†” (š‘¦ āˆˆ š“ āˆ§ š‘„ = āŸØš‘¦, šµāŸ©)))
4443mobidv 2062 . . . . 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©)))
455, 44mpbiri 168 . . . 4 (šµ āˆˆ š‘‰ ā†’ āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
4645alrimiv 1874 . . 3 (šµ āˆˆ š‘‰ ā†’ āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
47 funcnv2 5278 . . 3 (Fun ā—”(1st ā†¾ (š“ Ɨ {šµ})) ā†” āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ā†¾ (š“ Ɨ {šµ}))š‘¦)
4846, 47sylibr 134 . 2 (šµ āˆˆ š‘‰ ā†’ Fun ā—”(1st ā†¾ (š“ Ɨ {šµ})))
49 dff1o3 5469 . 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}))))
503, 48, 49sylanbrc 417 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 set class
Syntax hints:   ā†’ wi 4   āˆ§ wa 104   ā†” wb 105  āˆ€wal 1351   = wceq 1353  āˆƒwex 1492  āˆƒ*wmo 2027   āˆˆ wcel 2148  Vcvv 2739  {csn 3594  āŸØcop 3597   class class class wbr 4005   Ɨ cxp 4626  ā—”ccnv 4627   ā†¾ cres 4630  Fun wfun 5212   Fn wfn 5213  ā€“ontoā†’wfo 5216  ā€“1-1-ontoā†’wf1o 5217  ā€˜cfv 5218  1st c1st 6141  2nd c2nd 6142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144
This theorem is referenced by: (None)
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