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Theorem 2ndconst 6240
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
2ndconst (š“ ∈ š‘‰ → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–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 snmg 3724 . . 3 (š“ ∈ š‘‰ → āˆƒš‘„ š‘„ ∈ {š“})
2 fo2ndresm 6180 . . 3 (āˆƒš‘„ š‘„ ∈ {š“} → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–ontoā†’šµ)
31, 2syl 14 . 2 (š“ ∈ š‘‰ → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–ontoā†’šµ)
4 moeq 2926 . . . . . 6 ∃*š‘„ š‘„ = āŸØš“, š‘¦āŸ©
54moani 2107 . . . . 5 ∃*š‘„(š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)
6 vex 2754 . . . . . . . 8 š‘¦ ∈ V
76brres 4927 . . . . . . 7 (š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ (š‘„2nd š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)))
8 fo2nd 6176 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5454 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 2754 . . . . . . . . . 10 š‘„ ∈ V
12 fnbrfvb 5571 . . . . . . . . . 10 ((2nd Fn V ∧ š‘„ ∈ V) → ((2nd ā€˜š‘„) = š‘¦ ↔ š‘„2nd š‘¦))
1310, 11, 12mp2an 426 . . . . . . . . 9 ((2nd ā€˜š‘„) = š‘¦ ↔ š‘„2nd š‘¦)
1413anbi1i 458 . . . . . . . 8 (((2nd ā€˜š‘„) = š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)) ↔ (š‘„2nd š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)))
15 elxp7 6188 . . . . . . . . . . 11 (š‘„ ∈ ({š“} Ɨ šµ) ↔ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)))
16 eleq1 2251 . . . . . . . . . . . . . . 15 ((2nd ā€˜š‘„) = š‘¦ → ((2nd ā€˜š‘„) ∈ šµ ↔ š‘¦ ∈ šµ))
1716biimpa 296 . . . . . . . . . . . . . 14 (((2nd ā€˜š‘„) = š‘¦ ∧ (2nd ā€˜š‘„) ∈ šµ) → š‘¦ ∈ šµ)
1817adantrl 478 . . . . . . . . . . . . 13 (((2nd ā€˜š‘„) = š‘¦ ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)) → š‘¦ ∈ šµ)
1918adantrl 478 . . . . . . . . . . . 12 (((2nd ā€˜š‘„) = š‘¦ ∧ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ))) → š‘¦ ∈ šµ)
20 elsni 3624 . . . . . . . . . . . . . 14 ((1st ā€˜š‘„) ∈ {š“} → (1st ā€˜š‘„) = š“)
21 eqopi 6190 . . . . . . . . . . . . . . . 16 ((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) = š“ ∧ (2nd ā€˜š‘„) = š‘¦)) → š‘„ = āŸØš“, š‘¦āŸ©)
2221ancom2s 566 . . . . . . . . . . . . . . 15 ((š‘„ ∈ (V Ɨ V) ∧ ((2nd ā€˜š‘„) = š‘¦ ∧ (1st ā€˜š‘„) = š“)) → š‘„ = āŸØš“, š‘¦āŸ©)
2322an12s 565 . . . . . . . . . . . . . 14 (((2nd ā€˜š‘„) = š‘¦ ∧ (š‘„ ∈ (V Ɨ V) ∧ (1st ā€˜š‘„) = š“)) → š‘„ = āŸØš“, š‘¦āŸ©)
2420, 23sylanr2 405 . . . . . . . . . . . . 13 (((2nd ā€˜š‘„) = š‘¦ ∧ (š‘„ ∈ (V Ɨ V) ∧ (1st ā€˜š‘„) ∈ {š“})) → š‘„ = āŸØš“, š‘¦āŸ©)
2524adantrrr 487 . . . . . . . . . . . 12 (((2nd ā€˜š‘„) = š‘¦ ∧ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ))) → š‘„ = āŸØš“, š‘¦āŸ©)
2619, 25jca 306 . . . . . . . . . . 11 (((2nd ā€˜š‘„) = š‘¦ ∧ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ))) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
2715, 26sylan2b 287 . . . . . . . . . 10 (((2nd ā€˜š‘„) = š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
2827adantl 277 . . . . . . . . 9 ((š“ ∈ š‘‰ ∧ ((2nd ā€˜š‘„) = š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ))) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
29 fveq2 5529 . . . . . . . . . . . 12 (š‘„ = āŸØš“, š‘¦āŸ© → (2nd ā€˜š‘„) = (2nd ā€˜āŸØš“, š‘¦āŸ©))
30 op2ndg 6169 . . . . . . . . . . . . 13 ((š“ ∈ š‘‰ ∧ š‘¦ ∈ V) → (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
316, 30mpan2 425 . . . . . . . . . . . 12 (š“ ∈ š‘‰ → (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
3229, 31sylan9eqr 2243 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ š‘„ = āŸØš“, š‘¦āŸ©) → (2nd ā€˜š‘„) = š‘¦)
3332adantrl 478 . . . . . . . . . 10 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → (2nd ā€˜š‘„) = š‘¦)
34 simprr 531 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘„ = āŸØš“, š‘¦āŸ©)
35 snidg 3635 . . . . . . . . . . . . 13 (š“ ∈ š‘‰ → š“ ∈ {š“})
3635adantr 276 . . . . . . . . . . . 12 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š“ ∈ {š“})
37 simprl 529 . . . . . . . . . . . 12 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘¦ ∈ šµ)
38 opelxpi 4672 . . . . . . . . . . . 12 ((š“ ∈ {š“} ∧ š‘¦ ∈ šµ) → āŸØš“, š‘¦āŸ© ∈ ({š“} Ɨ šµ))
3936, 37, 38syl2anc 411 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → āŸØš“, š‘¦āŸ© ∈ ({š“} Ɨ šµ))
4034, 39eqeltrd 2265 . . . . . . . . . 10 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘„ ∈ ({š“} Ɨ šµ))
4133, 40jca 306 . . . . . . . . 9 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → ((2nd ā€˜š‘„) = š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)))
4228, 41impbida 596 . . . . . . . 8 (š“ ∈ š‘‰ → (((2nd ā€˜š‘„) = š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)) ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
4314, 42bitr3id 194 . . . . . . 7 (š“ ∈ š‘‰ → ((š‘„2nd š‘¦ ∧ š‘„ ∈ ({š“} Ɨ šµ)) ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
447, 43bitrid 192 . . . . . 6 (š“ ∈ š‘‰ → (š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
4544mobidv 2073 . . . . 5 (š“ ∈ š‘‰ → (∃*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ ∃*š‘„(š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
465, 45mpbiri 168 . . . 4 (š“ ∈ š‘‰ → ∃*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
4746alrimiv 1884 . . 3 (š“ ∈ š‘‰ → āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
48 funcnv2 5290 . . 3 (Fun ā—”(2nd ↾ ({š“} Ɨ šµ)) ↔ āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
4947, 48sylibr 134 . 2 (š“ ∈ š‘‰ → Fun ā—”(2nd ↾ ({š“} Ɨ šµ)))
50 dff1o3 5481 . 2 ((2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–1-1-ontoā†’šµ ↔ ((2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–ontoā†’šµ ∧ Fun ā—”(2nd ↾ ({š“} Ɨ šµ))))
513, 49, 50sylanbrc 417 1 (š“ ∈ š‘‰ → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–1-1-ontoā†’šµ)
Colors of variables: wff set class
Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  āˆ€wal 1361   = wceq 1363  āˆƒwex 1502  āˆƒ*wmo 2038   ∈ wcel 2159  Vcvv 2751  {csn 3606  āŸØcop 3609   class class class wbr 4017   Ɨ cxp 4638  ā—”ccnv 4639   ↾ cres 4642  Fun wfun 5224   Fn wfn 5225  ā€“onto→wfo 5228  ā€“1-1-onto→wf1o 5229  ā€˜cfv 5230  1st c1st 6156  2nd c2nd 6157
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-1st 6158  df-2nd 6159
This theorem is referenced by:  xpfi  6946  fsum2dlemstep  11459  fprod2dlemstep  11647
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