MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stconst Structured version   Visualization version   GIF version

Theorem 1stconst 7791
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 4709 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 7711 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3702 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2635 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3503 . . . . . . . 8 𝑦 ∈ V
76brresi 5861 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8 fo1st 7705 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6591 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3503 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6717 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 688 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi2i 622 . . . . . . . 8 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15 elxp7 7720 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2905 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpac 479 . . . . . . . . . . . . . 14 (((1st𝑥) ∈ 𝐴 ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1817adantlr 711 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1918adantll 710 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
20 elsni 4581 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 7721 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221anass1rs 651 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanl2 677 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantlrl 716 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 512 . . . . . . . . . . 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 482 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦)) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 769 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29 simprl 767 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
30 snidg 4596 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 481 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
3229, 31opelxpd 5592 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3328, 32eqeltrd 2918 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
3428fveq2d 6673 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35 simpl 483 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
36 op1stg 7697 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3729, 35, 36syl2anc 584 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3834, 37eqtrd 2861 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
3933, 38jca 512 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦))
4027, 39impbida 797 . . . . . . . 8 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4114, 40syl5bbr 286 . . . . . . 7 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
427, 41syl5bb 284 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4342mobidv 2631 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
445, 43mpbiri 259 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4544alrimiv 1921 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46 funcnv2 6421 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4745, 46sylibr 235 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
48 dff1o3 6620 . 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 207  wa 396  wal 1528   = wceq 1530  wcel 2107  ∃*wmo 2618  wne 3021  Vcvv 3500  c0 4295  {csn 4564  cop 4570   class class class wbr 5063   × cxp 5552  ccnv 5553  cres 5556  Fun wfun 6348   Fn wfn 6349  ontowfo 6352  1-1-ontowf1o 6353  cfv 6354  1st c1st 7683  2nd c2nd 7684
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-1st 7685  df-2nd 7686
This theorem is referenced by:  curry2  7798  domss2  8670  fv1stcnv  32923
  Copyright terms: Public domain W3C validator