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Theorem 2ndconst 7434
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
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 4451 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 7360 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3523 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2663 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3343 . . . . . . . 8 𝑦 ∈ V
76brres 5560 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
8 fo2nd 7354 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6278 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3343 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6397 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 710 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi1i 733 . . . . . . . 8 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
15 elxp7 7368 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2827 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpa 502 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (2nd𝑥) ∈ 𝐵) → 𝑦𝐵)
1817adantrl 754 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) → 𝑦𝐵)
1918adantrl 754 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑦𝐵)
20 elsni 4338 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 7369 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221ancom2s 879 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((2nd𝑥) = 𝑦 ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2322an12s 878 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2420, 23sylanr2 688 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
2524adantrrr 763 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
2619, 25jca 555 . . . . . . . . . . 11 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2715, 26sylan2b 493 . . . . . . . . . 10 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2827adantl 473 . . . . . . . . 9 ((𝐴𝑉 ∧ ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
29 fveq2 6352 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30 op2ndg 7346 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
316, 30mpan2 709 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3229, 31sylan9eqr 2816 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3332adantrl 754 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
34 simprr 813 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35 snidg 4351 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3635adantr 472 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37 simprl 811 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
38 opelxpi 5305 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3936, 37, 38syl2anc 696 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
4034, 39eqeltrd 2839 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
4133, 40jca 555 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
4228, 41impbida 913 . . . . . . . 8 (𝐴𝑉 → (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4314, 42syl5bbr 274 . . . . . . 7 (𝐴𝑉 → ((𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
447, 43syl5bb 272 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4544mobidv 2628 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
465, 45mpbiri 248 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4746alrimiv 2004 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48 funcnv2 6118 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4947, 48sylibr 224 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
50 dff1o3 6304 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
513, 49, 50sylanbrc 701 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  ∃*wmo 2608  wne 2932  Vcvv 3340  c0 4058  {csn 4321  cop 4327   class class class wbr 4804   × cxp 5264  ccnv 5265  cres 5268  Fun wfun 6043   Fn wfn 6044  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  1st c1st 7331  2nd c2nd 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-1st 7333  df-2nd 7334
This theorem is referenced by:  curry1  7437  xpfi  8396  fsum2dlem  14700  fprod2dlem  14909  gsum2dlem2  18570  ovoliunlem1  23470  gsummpt2d  30090  fv2ndcnv  31986
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