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Theorem 1stconst 8101
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 ∈ š‘‰ → (1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–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 4772 . . 3 (šµ ∈ š‘‰ → {šµ} ≠ āˆ…)
2 fo1stres 8015 . . 3 ({šµ} ≠ āˆ… → (1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–ontoā†’š“)
31, 2syl 17 . 2 (šµ ∈ š‘‰ → (1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–ontoā†’š“)
4 moeq 3694 . . . . . 6 ∃*š‘„ š‘„ = āŸØš‘¦, šµāŸ©
54moani 2541 . . . . 5 ∃*š‘„(š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)
6 vex 3467 . . . . . . . 8 š‘¦ ∈ V
76brresi 5986 . . . . . . 7 (š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦ ↔ (š‘„ ∈ (š“ Ɨ {šµ}) ∧ š‘„1st š‘¦))
8 fo1st 8009 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6806 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3467 . . . . . . . . . 10 š‘„ ∈ V
12 fnbrfvb 6943 . . . . . . . . . 10 ((1st Fn V ∧ š‘„ ∈ V) → ((1st ā€˜š‘„) = š‘¦ ↔ š‘„1st š‘¦))
1310, 11, 12mp2an 690 . . . . . . . . 9 ((1st ā€˜š‘„) = š‘¦ ↔ š‘„1st š‘¦)
1413anbi2i 621 . . . . . . . 8 ((š‘„ ∈ (š“ Ɨ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦) ↔ (š‘„ ∈ (š“ Ɨ {šµ}) ∧ š‘„1st š‘¦))
15 elxp7 8024 . . . . . . . . . . 11 (š‘„ ∈ (š“ Ɨ {šµ}) ↔ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ {šµ})))
16 eleq1 2813 . . . . . . . . . . . . . . 15 ((1st ā€˜š‘„) = š‘¦ → ((1st ā€˜š‘„) ∈ š“ ↔ š‘¦ ∈ š“))
1716biimpac 477 . . . . . . . . . . . . . 14 (((1st ā€˜š‘„) ∈ š“ ∧ (1st ā€˜š‘„) = š‘¦) → š‘¦ ∈ š“)
1817adantlr 713 . . . . . . . . . . . . 13 ((((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦) → š‘¦ ∈ š“)
1918adantll 712 . . . . . . . . . . . 12 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ {šµ})) ∧ (1st ā€˜š‘„) = š‘¦) → š‘¦ ∈ š“)
20 elsni 4639 . . . . . . . . . . . . . 14 ((2nd ā€˜š‘„) ∈ {šµ} → (2nd ā€˜š‘„) = šµ)
21 eqopi 8025 . . . . . . . . . . . . . . 15 ((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) = š‘¦ ∧ (2nd ā€˜š‘„) = šµ)) → š‘„ = āŸØš‘¦, šµāŸ©)
2221anass1rs 653 . . . . . . . . . . . . . 14 (((š‘„ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘„) = šµ) ∧ (1st ā€˜š‘„) = š‘¦) → š‘„ = āŸØš‘¦, šµāŸ©)
2320, 22sylanl2 679 . . . . . . . . . . . . 13 (((š‘„ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘„) ∈ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦) → š‘„ = āŸØš‘¦, šµāŸ©)
2423adantlrl 718 . . . . . . . . . . . 12 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ {šµ})) ∧ (1st ā€˜š‘„) = š‘¦) → š‘„ = āŸØš‘¦, šµāŸ©)
2519, 24jca 510 . . . . . . . . . . 11 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ š“ ∧ (2nd ā€˜š‘„) ∈ {šµ})) ∧ (1st ā€˜š‘„) = š‘¦) → (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©))
2615, 25sylanb 579 . . . . . . . . . 10 ((š‘„ ∈ (š“ Ɨ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦) → (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©))
2726adantl 480 . . . . . . . . 9 ((šµ ∈ š‘‰ ∧ (š‘„ ∈ (š“ Ɨ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦)) → (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©))
28 simprr 771 . . . . . . . . . . 11 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → š‘„ = āŸØš‘¦, šµāŸ©)
29 simprl 769 . . . . . . . . . . . 12 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → š‘¦ ∈ š“)
30 snidg 4656 . . . . . . . . . . . . 13 (šµ ∈ š‘‰ → šµ ∈ {šµ})
3130adantr 479 . . . . . . . . . . . 12 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → šµ ∈ {šµ})
3229, 31opelxpd 5709 . . . . . . . . . . 11 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → āŸØš‘¦, šµāŸ© ∈ (š“ Ɨ {šµ}))
3328, 32eqeltrd 2825 . . . . . . . . . 10 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → š‘„ ∈ (š“ Ɨ {šµ}))
3428fveq2d 6894 . . . . . . . . . . 11 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → (1st ā€˜š‘„) = (1st ā€˜āŸØš‘¦, šµāŸ©))
35 simpl 481 . . . . . . . . . . . 12 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → šµ ∈ š‘‰)
36 op1stg 8001 . . . . . . . . . . . 12 ((š‘¦ ∈ š“ ∧ šµ ∈ š‘‰) → (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3729, 35, 36syl2anc 582 . . . . . . . . . . 11 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → (1st ā€˜āŸØš‘¦, šµāŸ©) = š‘¦)
3834, 37eqtrd 2765 . . . . . . . . . 10 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → (1st ā€˜š‘„) = š‘¦)
3933, 38jca 510 . . . . . . . . 9 ((šµ ∈ š‘‰ ∧ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)) → (š‘„ ∈ (š“ Ɨ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦))
4027, 39impbida 799 . . . . . . . 8 (šµ ∈ š‘‰ → ((š‘„ ∈ (š“ Ɨ {šµ}) ∧ (1st ā€˜š‘„) = š‘¦) ↔ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)))
4114, 40bitr3id 284 . . . . . . 7 (šµ ∈ š‘‰ → ((š‘„ ∈ (š“ Ɨ {šµ}) ∧ š‘„1st š‘¦) ↔ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)))
427, 41bitrid 282 . . . . . 6 (šµ ∈ š‘‰ → (š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦ ↔ (š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)))
4342mobidv 2537 . . . . 5 (šµ ∈ š‘‰ → (∃*š‘„ š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦ ↔ ∃*š‘„(š‘¦ ∈ š“ ∧ š‘„ = āŸØš‘¦, šµāŸ©)))
445, 43mpbiri 257 . . . 4 (šµ ∈ š‘‰ → ∃*š‘„ š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦)
4544alrimiv 1922 . . 3 (šµ ∈ š‘‰ → āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦)
46 funcnv2 6614 . . 3 (Fun ā—”(1st ↾ (š“ Ɨ {šµ})) ↔ āˆ€š‘¦āˆƒ*š‘„ š‘„(1st ↾ (š“ Ɨ {šµ}))š‘¦)
4745, 46sylibr 233 . 2 (šµ ∈ š‘‰ → Fun ā—”(1st ↾ (š“ Ɨ {šµ})))
48 dff1o3 6838 . 2 ((1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–1-1-ontoā†’š“ ↔ ((1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–ontoā†’š“ ∧ Fun ā—”(1st ↾ (š“ Ɨ {šµ}))))
493, 47, 48sylanbrc 581 1 (šµ ∈ š‘‰ → (1st ↾ (š“ Ɨ {šµ})):(š“ Ɨ {šµ})–1-1-ontoā†’š“)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  āˆ€wal 1531   = wceq 1533   ∈ wcel 2098  āˆƒ*wmo 2526   ≠ wne 2930  Vcvv 3463  āˆ…c0 4316  {csn 4622  āŸØcop 4628   class class class wbr 5141   Ɨ cxp 5668  ā—”ccnv 5669   ↾ cres 5672  Fun wfun 6535   Fn wfn 6536  ā€“onto→wfo 6539  ā€“1-1-onto→wf1o 6540  ā€˜cfv 6541  1st c1st 7987  2nd c2nd 7988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-1st 7989  df-2nd 7990
This theorem is referenced by:  curry2  8108  domss2  9157  fv1stcnv  35401
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