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Theorem 1stconst 8124
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 (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 4779 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 8039 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3716 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2551 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3482 . . . . . . . 8 𝑦 ∈ V
76brresi 6009 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8 fo1st 8033 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6823 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3482 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6960 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 692 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi2i 623 . . . . . . . 8 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15 elxp7 8048 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2827 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpac 478 . . . . . . . . . . . . . 14 (((1st𝑥) ∈ 𝐴 ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1817adantlr 715 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1918adantll 714 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
20 elsni 4648 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 8049 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221anass1rs 655 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanl2 681 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantlrl 720 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 511 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylanb 581 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 481 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦)) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 773 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29 simprl 771 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
30 snidg 4665 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 480 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
3229, 31opelxpd 5728 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3328, 32eqeltrd 2839 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
3428fveq2d 6911 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35 simpl 482 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
36 op1stg 8025 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3729, 35, 36syl2anc 584 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3834, 37eqtrd 2775 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
3933, 38jca 511 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦))
4027, 39impbida 801 . . . . . . . 8 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4114, 40bitr3id 285 . . . . . . 7 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
427, 41bitrid 283 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4342mobidv 2547 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
445, 43mpbiri 258 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4544alrimiv 1925 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46 funcnv2 6636 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4745, 46sylibr 234 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
48 dff1o3 6855 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
493, 47, 48sylanbrc 583 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  wne 2938  Vcvv 3478  c0 4339  {csn 4631  cop 4637   class class class wbr 5148   × cxp 5687  ccnv 5688  cres 5691  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  curry2  8131  domss2  9175  fv1stcnv  35758
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