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Theorem 2ndconst 8110
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
2ndconst (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)

Proof of Theorem 2ndconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnzg 4781 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 8024 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3702 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2542 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3475 . . . . . . . 8 𝑦 ∈ V
76brresi 5996 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8 fo2nd 8018 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6816 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3475 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6953 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 690 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi2i 621 . . . . . . . 8 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15 elxp7 8032 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2816 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpac 477 . . . . . . . . . . . . . 14 (((2nd𝑥) ∈ 𝐵 ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1817adantll 712 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1918adantll 712 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
20 elsni 4647 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 8033 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221anassrs 466 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2320, 22sylanl2 679 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴}) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2423adantlrr 719 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2519, 24jca 510 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2615, 25sylanb 579 . . . . . . . . . 10 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2726adantl 480 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
28 simprr 771 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29 snidg 4665 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3029adantr 479 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31 simprl 769 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
3230, 31opelxpd 5719 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3328, 32eqeltrd 2828 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34 fveq2 6900 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35 op2ndg 8010 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3635elvd 3478 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3734, 36sylan9eqr 2789 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3837adantrl 714 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
3933, 38jca 510 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦))
4027, 39impbida 799 . . . . . . . 8 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4114, 40bitr3id 284 . . . . . . 7 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
427, 41bitrid 282 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4342mobidv 2538 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
445, 43mpbiri 257 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4544alrimiv 1922 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46 funcnv2 6624 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4745, 46sylibr 233 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
48 dff1o3 6848 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
493, 47, 48sylanbrc 581 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wcel 2098  ∃*wmo 2527  wne 2936  Vcvv 3471  c0 4324  {csn 4630  cop 4636   class class class wbr 5150   × cxp 5678  ccnv 5679  cres 5682  Fun wfun 6545   Fn wfn 6546  ontowfo 6549  1-1-ontowf1o 6550  cfv 6551  1st c1st 7995  2nd c2nd 7996
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-1st 7997  df-2nd 7998
This theorem is referenced by:  curry1  8113  xpfiOLD  9348  fsum2dlem  15754  fprod2dlem  15962  gsum2dlem2  19931  ovoliunlem1  25449  gsummpt2d  32781  fv2ndcnv  35378
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