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Theorem 1stconst 7493
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 snnzg 4492 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 7418 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3574 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2686 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3390 . . . . . . . 8 𝑦 ∈ V
76brres 5600 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
8 fo1st 7412 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6327 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3390 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6450 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 675 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi1i 612 . . . . . . . 8 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
15 elxp7 7427 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2869 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpa 464 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (1st𝑥) ∈ 𝐴) → 𝑦𝐴)
1817adantrr 699 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) → 𝑦𝐴)
1918adantrl 698 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑦𝐴)
20 elsni 4381 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 7428 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221an12s 631 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanr2 665 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantrrl 706 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 503 . . . . . . . . . . 11 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylan2b 583 . . . . . . . . . 10 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 469 . . . . . . . . 9 ((𝐵𝑉 ∧ ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 780 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2928fveq2d 6406 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30 simprl 778 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
31 simpl 470 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
32 op1stg 7404 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3330, 31, 32syl2anc 575 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3429, 33eqtrd 2836 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
35 snidg 4394 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3635adantr 468 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37 opelxpi 5342 . . . . . . . . . . . 12 ((𝑦𝐴𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3830, 36, 37syl2anc 575 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3928, 38eqeltrd 2881 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
4034, 39jca 503 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})))
4127, 40impbida 826 . . . . . . . 8 (𝐵𝑉 → (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4214, 41syl5bbr 276 . . . . . . 7 (𝐵𝑉 → ((𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
437, 42syl5bb 274 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4443mobidv 2650 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
455, 44mpbiri 249 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4645alrimiv 2017 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47 funcnv2 6162 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4846, 47sylibr 225 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
49 dff1o3 6353 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
503, 48, 49sylanbrc 574 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wcel 2155  ∃*wmo 2630  wne 2974  Vcvv 3387  c0 4110  {csn 4364  cop 4370   class class class wbr 4837   × cxp 5303  ccnv 5304  cres 5307  Fun wfun 6089   Fn wfn 6090  ontowfo 6093  1-1-ontowf1o 6094  cfv 6095  1st c1st 7390  2nd c2nd 7391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-1st 7392  df-2nd 7393
This theorem is referenced by:  curry2  7500  domss2  8352  fv1stcnv  31994
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