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Theorem 1stconst 5968
Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
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 snmg 3553 . . 3 (𝐵𝑉 → ∃𝑥 𝑥 ∈ {𝐵})
2 fo1stresm 5914 . . 3 (∃𝑥 𝑥 ∈ {𝐵} → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 14 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 2788 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2018 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 2622 . . . . . . . 8 𝑦 ∈ V
76brres 4707 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
8 fo1st 5910 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 5219 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 7 . . . . . . . . . 10 1st Fn V
11 vex 2622 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5329 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 417 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi1i 446 . . . . . . . 8 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
15 elxp7 5923 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2150 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpa 290 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (1st𝑥) ∈ 𝐴) → 𝑦𝐴)
1817adantrr 463 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) → 𝑦𝐴)
1918adantrl 462 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑦𝐴)
20 elsni 3459 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 5924 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221an12s 532 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanr2 397 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantrrl 470 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 300 . . . . . . . . . . 11 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylan2b 281 . . . . . . . . . 10 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 271 . . . . . . . . 9 ((𝐵𝑉 ∧ ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 499 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2928fveq2d 5293 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30 simprl 498 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
31 simpl 107 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
32 op1stg 5903 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3330, 31, 32syl2anc 403 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3429, 33eqtrd 2120 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
35 snidg 3468 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3635adantr 270 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37 opelxpi 4459 . . . . . . . . . . . 12 ((𝑦𝐴𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3830, 36, 37syl2anc 403 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3928, 38eqeltrd 2164 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
4034, 39jca 300 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})))
4127, 40impbida 563 . . . . . . . 8 (𝐵𝑉 → (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4214, 41syl5bbr 192 . . . . . . 7 (𝐵𝑉 → ((𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
437, 42syl5bb 190 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4443mobidv 1984 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
455, 44mpbiri 166 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4645alrimiv 1802 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47 funcnv2 5060 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4846, 47sylibr 132 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
49 dff1o3 5243 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
503, 48, 49sylanbrc 408 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃*wmo 1949  Vcvv 2619  {csn 3441  cop 3444   class class class wbr 3837   × cxp 4426  ccnv 4427  cres 4430  Fun wfun 4996   Fn wfn 4997  ontowfo 5000  1-1-ontowf1o 5001  cfv 5002  1st c1st 5891  2nd c2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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