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Theorem 2ndconst 7801
 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 4667 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 7720 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3621 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2571 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3413 . . . . . . . 8 𝑦 ∈ V
76brresi 5832 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8 fo2nd 7714 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6578 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3413 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6706 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi2i 625 . . . . . . . 8 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15 elxp7 7728 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2839 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpac 482 . . . . . . . . . . . . . 14 (((2nd𝑥) ∈ 𝐵 ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1817adantll 713 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1918adantll 713 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
20 elsni 4539 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 7729 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221anassrs 471 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴}) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2423adantlrr 720 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2519, 24jca 515 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2615, 25sylanb 584 . . . . . . . . . 10 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2726adantl 485 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
28 simprr 772 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29 snidg 4556 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3029adantr 484 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31 simprl 770 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
3230, 31opelxpd 5562 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3328, 32eqeltrd 2852 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34 fveq2 6658 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35 op2ndg 7706 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3635elvd 3416 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3734, 36sylan9eqr 2815 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3837adantrl 715 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
3933, 38jca 515 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦))
4027, 39impbida 800 . . . . . . . 8 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4114, 40bitr3id 288 . . . . . . 7 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
427, 41syl5bb 286 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4342mobidv 2567 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
445, 43mpbiri 261 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4544alrimiv 1928 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46 funcnv2 6403 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4745, 46sylibr 237 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
48 dff1o3 6608 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
493, 47, 48sylanbrc 586 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2555   ≠ wne 2951  Vcvv 3409  ∅c0 4225  {csn 4522  ⟨cop 4528   class class class wbr 5032   × cxp 5522  ◡ccnv 5523   ↾ cres 5526  Fun wfun 6329   Fn wfn 6330  –onto→wfo 6333  –1-1-onto→wf1o 6334  ‘cfv 6335  1st c1st 7691  2nd c2nd 7692 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 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-1st 7693  df-2nd 7694 This theorem is referenced by:  curry1  7804  xpfi  8822  fsum2dlem  15173  fprod2dlem  15382  gsum2dlem2  19159  ovoliunlem1  24202  gsummpt2d  30835  fv2ndcnv  33268
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