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Theorem 1stconst 7785
 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 4670 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 7704 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3646 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2612 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3444 . . . . . . . 8 𝑦 ∈ V
76brresi 5828 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8 fo1st 7698 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6570 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3444 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6698 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi2i 625 . . . . . . . 8 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15 elxp7 7713 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2877 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpac 482 . . . . . . . . . . . . . 14 (((1st𝑥) ∈ 𝐴 ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1817adantlr 714 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1918adantll 713 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
20 elsni 4542 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 7714 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221anass1rs 654 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantlrl 719 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 515 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylanb 584 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 485 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦)) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 772 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29 simprl 770 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
30 snidg 4559 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 484 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
3229, 31opelxpd 5558 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3328, 32eqeltrd 2890 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
3428fveq2d 6654 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35 simpl 486 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
36 op1stg 7690 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3729, 35, 36syl2anc 587 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3834, 37eqtrd 2833 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
3933, 38jca 515 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦))
4027, 39impbida 800 . . . . . . . 8 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4114, 40bitr3id 288 . . . . . . 7 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
427, 41syl5bb 286 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4342mobidv 2608 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
445, 43mpbiri 261 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4544alrimiv 1928 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46 funcnv2 6395 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4745, 46sylibr 237 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
48 dff1o3 6600 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
493, 47, 48sylanbrc 586 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596   ≠ wne 2987  Vcvv 3441  ∅c0 4243  {csn 4525  ⟨cop 4531   class class class wbr 5031   × cxp 5518  ◡ccnv 5519   ↾ cres 5522  Fun wfun 6321   Fn wfn 6322  –onto→wfo 6325  –1-1-onto→wf1o 6326  ‘cfv 6327  1st c1st 7676  2nd c2nd 7677 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-1st 7678  df-2nd 7679 This theorem is referenced by:  curry2  7792  domss2  8667  fv1stcnv  33169
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