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Theorem 1stconst 8125
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 4774 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 8040 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3713 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2553 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3484 . . . . . . . 8 𝑦 ∈ V
76brresi 6006 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8 fo1st 8034 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6822 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6959 . . . . . . . . . 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 8049 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2829 . . . . . . . . . . . . . . 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 4643 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 8050 . . . . . . . . . . . . . . 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 4660 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 480 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
3229, 31opelxpd 5724 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3328, 32eqeltrd 2841 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
3428fveq2d 6910 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35 simpl 482 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
36 op1stg 8026 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3729, 35, 36syl2anc 584 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3834, 37eqtrd 2777 . . . . . . . . . 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 2549 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
445, 43mpbiri 258 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4544alrimiv 1927 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46 funcnv2 6634 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4745, 46sylibr 234 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
48 dff1o3 6854 . 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 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wne 2940  Vcvv 3480  c0 4333  {csn 4626  cop 4632   class class class wbr 5143   × cxp 5683  ccnv 5684  cres 5687  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  curry2  8132  domss2  9176  fv1stcnv  35777
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