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Theorem 2ndconst 8033
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 4735 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 7948 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3665 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2551 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3449 . . . . . . . 8 𝑦 ∈ V
76brresi 5946 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8 fo2nd 7942 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6758 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6895 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 690 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi2i 623 . . . . . . . 8 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15 elxp7 7956 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2825 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpac 479 . . . . . . . . . . . . . 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 4603 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 7957 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221anassrs 468 . . . . . . . . . . . . . 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 512 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2615, 25sylanb 581 . . . . . . . . . 10 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2726adantl 482 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
28 simprr 771 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29 snidg 4620 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3029adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31 simprl 769 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
3230, 31opelxpd 5671 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3328, 32eqeltrd 2838 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35 op2ndg 7934 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3635elvd 3452 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3734, 36sylan9eqr 2798 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3837adantrl 714 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
3933, 38jca 512 . . . . . . . . 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 2547 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
445, 43mpbiri 257 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4544alrimiv 1930 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46 funcnv2 6569 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4745, 46sylibr 233 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
48 dff1o3 6790 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
493, 47, 48sylanbrc 583 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2536  wne 2943  Vcvv 3445  c0 4282  {csn 4586  cop 4592   class class class wbr 5105   × cxp 5631  ccnv 5632  cres 5635  Fun wfun 6490   Fn wfn 6491  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  1st c1st 7919  2nd c2nd 7920
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1st 7921  df-2nd 7922
This theorem is referenced by:  curry1  8036  xpfiOLD  9262  fsum2dlem  15655  fprod2dlem  15863  gsum2dlem2  19748  ovoliunlem1  24866  gsummpt2d  31891  fv2ndcnv  34352
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