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Theorem 2ndconst 8100
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“ ∈ 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 snnzg 4774 . . 3 (š“ ∈ š‘‰ → {š“} ≠ āˆ…)
2 fo2ndres 8014 . . 3 ({š“} ≠ āˆ… → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–ontoā†’šµ)
31, 2syl 17 . 2 (š“ ∈ š‘‰ → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–ontoā†’šµ)
4 moeq 3700 . . . . . 6 ∃*š‘„ š‘„ = āŸØš“, š‘¦āŸ©
54moani 2542 . . . . 5 ∃*š‘„(š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)
6 vex 3473 . . . . . . . 8 š‘¦ ∈ V
76brresi 5988 . . . . . . 7 (š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ (š‘„ ∈ ({š“} Ɨ šµ) ∧ š‘„2nd š‘¦))
8 fo2nd 8008 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6807 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3473 . . . . . . . . . 10 š‘„ ∈ V
12 fnbrfvb 6944 . . . . . . . . . 10 ((2nd Fn V ∧ š‘„ ∈ V) → ((2nd ā€˜š‘„) = š‘¦ ↔ š‘„2nd š‘¦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((2nd ā€˜š‘„) = š‘¦ ↔ š‘„2nd š‘¦)
1413anbi2i 622 . . . . . . . 8 ((š‘„ ∈ ({š“} Ɨ šµ) ∧ (2nd ā€˜š‘„) = š‘¦) ↔ (š‘„ ∈ ({š“} Ɨ šµ) ∧ š‘„2nd š‘¦))
15 elxp7 8022 . . . . . . . . . . 11 (š‘„ ∈ ({š“} Ɨ šµ) ↔ (š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)))
16 eleq1 2816 . . . . . . . . . . . . . . 15 ((2nd ā€˜š‘„) = š‘¦ → ((2nd ā€˜š‘„) ∈ šµ ↔ š‘¦ ∈ šµ))
1716biimpac 478 . . . . . . . . . . . . . 14 (((2nd ā€˜š‘„) ∈ šµ ∧ (2nd ā€˜š‘„) = š‘¦) → š‘¦ ∈ šµ)
1817adantll 713 . . . . . . . . . . . . 13 ((((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ) ∧ (2nd ā€˜š‘„) = š‘¦) → š‘¦ ∈ šµ)
1918adantll 713 . . . . . . . . . . . 12 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)) ∧ (2nd ā€˜š‘„) = š‘¦) → š‘¦ ∈ šµ)
20 elsni 4641 . . . . . . . . . . . . . 14 ((1st ā€˜š‘„) ∈ {š“} → (1st ā€˜š‘„) = š“)
21 eqopi 8023 . . . . . . . . . . . . . . 15 ((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) = š“ ∧ (2nd ā€˜š‘„) = š‘¦)) → š‘„ = āŸØš“, š‘¦āŸ©)
2221anassrs 467 . . . . . . . . . . . . . 14 (((š‘„ ∈ (V Ɨ V) ∧ (1st ā€˜š‘„) = š“) ∧ (2nd ā€˜š‘„) = š‘¦) → š‘„ = āŸØš“, š‘¦āŸ©)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((š‘„ ∈ (V Ɨ V) ∧ (1st ā€˜š‘„) ∈ {š“}) ∧ (2nd ā€˜š‘„) = š‘¦) → š‘„ = āŸØš“, š‘¦āŸ©)
2423adantlrr 720 . . . . . . . . . . . 12 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)) ∧ (2nd ā€˜š‘„) = š‘¦) → š‘„ = āŸØš“, š‘¦āŸ©)
2519, 24jca 511 . . . . . . . . . . 11 (((š‘„ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘„) ∈ {š“} ∧ (2nd ā€˜š‘„) ∈ šµ)) ∧ (2nd ā€˜š‘„) = š‘¦) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
2615, 25sylanb 580 . . . . . . . . . 10 ((š‘„ ∈ ({š“} Ɨ šµ) ∧ (2nd ā€˜š‘„) = š‘¦) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
2726adantl 481 . . . . . . . . 9 ((š“ ∈ š‘‰ ∧ (š‘„ ∈ ({š“} Ɨ šµ) ∧ (2nd ā€˜š‘„) = š‘¦)) → (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©))
28 simprr 772 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘„ = āŸØš“, š‘¦āŸ©)
29 snidg 4658 . . . . . . . . . . . . 13 (š“ ∈ š‘‰ → š“ ∈ {š“})
3029adantr 480 . . . . . . . . . . . 12 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š“ ∈ {š“})
31 simprl 770 . . . . . . . . . . . 12 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘¦ ∈ šµ)
3230, 31opelxpd 5711 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → āŸØš“, š‘¦āŸ© ∈ ({š“} Ɨ šµ))
3328, 32eqeltrd 2828 . . . . . . . . . 10 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → š‘„ ∈ ({š“} Ɨ šµ))
34 fveq2 6891 . . . . . . . . . . . 12 (š‘„ = āŸØš“, š‘¦āŸ© → (2nd ā€˜š‘„) = (2nd ā€˜āŸØš“, š‘¦āŸ©))
35 op2ndg 8000 . . . . . . . . . . . . 13 ((š“ ∈ š‘‰ ∧ š‘¦ ∈ V) → (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
3635elvd 3476 . . . . . . . . . . . 12 (š“ ∈ š‘‰ → (2nd ā€˜āŸØš“, š‘¦āŸ©) = š‘¦)
3734, 36sylan9eqr 2789 . . . . . . . . . . 11 ((š“ ∈ š‘‰ ∧ š‘„ = āŸØš“, š‘¦āŸ©) → (2nd ā€˜š‘„) = š‘¦)
3837adantrl 715 . . . . . . . . . 10 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → (2nd ā€˜š‘„) = š‘¦)
3933, 38jca 511 . . . . . . . . 9 ((š“ ∈ š‘‰ ∧ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)) → (š‘„ ∈ ({š“} Ɨ šµ) ∧ (2nd ā€˜š‘„) = š‘¦))
4027, 39impbida 800 . . . . . . . 8 (š“ ∈ š‘‰ → ((š‘„ ∈ ({š“} Ɨ šµ) ∧ (2nd ā€˜š‘„) = š‘¦) ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
4114, 40bitr3id 285 . . . . . . 7 (š“ ∈ š‘‰ → ((š‘„ ∈ ({š“} Ɨ šµ) ∧ š‘„2nd š‘¦) ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
427, 41bitrid 283 . . . . . 6 (š“ ∈ š‘‰ → (š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ (š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
4342mobidv 2538 . . . . 5 (š“ ∈ š‘‰ → (∃*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦ ↔ ∃*š‘„(š‘¦ ∈ šµ ∧ š‘„ = āŸØš“, š‘¦āŸ©)))
445, 43mpbiri 258 . . . 4 (š“ ∈ š‘‰ → ∃*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
4544alrimiv 1923 . . 3 (š“ ∈ š‘‰ → āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
46 funcnv2 6615 . . 3 (Fun ā—”(2nd ↾ ({š“} Ɨ šµ)) ↔ āˆ€š‘¦āˆƒ*š‘„ š‘„(2nd ↾ ({š“} Ɨ šµ))š‘¦)
4745, 46sylibr 233 . 2 (š“ ∈ š‘‰ → Fun ā—”(2nd ↾ ({š“} Ɨ šµ)))
48 dff1o3 6839 . 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))))
493, 47, 48sylanbrc 582 1 (š“ ∈ š‘‰ → (2nd ↾ ({š“} Ɨ šµ)):({š“} Ɨ šµ)–1-1-ontoā†’šµ)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  āˆ€wal 1532   = wceq 1534   ∈ wcel 2099  āˆƒ*wmo 2527   ≠ wne 2935  Vcvv 3469  āˆ…c0 4318  {csn 4624  āŸØcop 4630   class class class wbr 5142   Ɨ cxp 5670  ā—”ccnv 5671   ↾ cres 5674  Fun wfun 6536   Fn wfn 6537  ā€“onto→wfo 6540  ā€“1-1-onto→wf1o 6541  ā€˜cfv 6542  1st c1st 7985  2nd c2nd 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7987  df-2nd 7988
This theorem is referenced by:  curry1  8103  xpfiOLD  9334  fsum2dlem  15740  fprod2dlem  15948  gsum2dlem2  19917  ovoliunlem1  25418  gsummpt2d  32741  fv2ndcnv  35309
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