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Theorem 1stconst 6168
Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
1stconst (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)

Proof of Theorem 1stconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3677 . . 3 (𝐵𝑉 → ∃𝑥 𝑥 ∈ {𝐵})
2 fo1stresm 6109 . . 3 (∃𝑥 𝑥 ∈ {𝐵} → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 14 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 2887 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2076 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 2715 . . . . . . . 8 𝑦 ∈ V
76brres 4872 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
8 fo1st 6105 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 5394 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 2715 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5509 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 423 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi1i 454 . . . . . . . 8 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
15 elxp7 6118 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2220 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpa 294 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (1st𝑥) ∈ 𝐴) → 𝑦𝐴)
1817adantrr 471 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) → 𝑦𝐴)
1918adantrl 470 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑦𝐴)
20 elsni 3578 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 6120 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221an12s 555 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanr2 403 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantrrl 478 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 304 . . . . . . . . . . 11 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylan2b 285 . . . . . . . . . 10 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 275 . . . . . . . . 9 ((𝐵𝑉 ∧ ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 522 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2928fveq2d 5472 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30 simprl 521 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
31 simpl 108 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
32 op1stg 6098 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3330, 31, 32syl2anc 409 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3429, 33eqtrd 2190 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
35 snidg 3589 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3635adantr 274 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37 opelxpi 4618 . . . . . . . . . . . 12 ((𝑦𝐴𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3830, 36, 37syl2anc 409 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3928, 38eqeltrd 2234 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
4034, 39jca 304 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})))
4127, 40impbida 586 . . . . . . . 8 (𝐵𝑉 → (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4214, 41bitr3id 193 . . . . . . 7 (𝐵𝑉 → ((𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
437, 42syl5bb 191 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4443mobidv 2042 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
455, 44mpbiri 167 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4645alrimiv 1854 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47 funcnv2 5230 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4846, 47sylibr 133 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
49 dff1o3 5420 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
503, 48, 49sylanbrc 414 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333   = wceq 1335  wex 1472  ∃*wmo 2007  wcel 2128  Vcvv 2712  {csn 3560  cop 3563   class class class wbr 3965   × cxp 4584  ccnv 4585  cres 4588  Fun wfun 5164   Fn wfn 5165  ontowfo 5168  1-1-ontowf1o 5169  cfv 5170  1st c1st 6086  2nd c2nd 6087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089
This theorem is referenced by: (None)
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