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Theorem 1stconst 6216
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 3709 . . 3 (𝐵𝑉 → ∃𝑥 𝑥 ∈ {𝐵})
2 fo1stresm 6156 . . 3 (∃𝑥 𝑥 ∈ {𝐵} → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 14 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 2912 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2096 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 2740 . . . . . . . 8 𝑦 ∈ V
76brres 4909 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
8 fo1st 6152 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 5436 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 2740 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5552 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 426 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi1i 458 . . . . . . . 8 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
15 elxp7 6165 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2240 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpa 296 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (1st𝑥) ∈ 𝐴) → 𝑦𝐴)
1817adantrr 479 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) → 𝑦𝐴)
1918adantrl 478 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑦𝐴)
20 elsni 3609 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 6167 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221an12s 565 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanr2 405 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantrrl 486 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 306 . . . . . . . . . . 11 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylan2b 287 . . . . . . . . . 10 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 277 . . . . . . . . 9 ((𝐵𝑉 ∧ ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 531 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2928fveq2d 5515 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30 simprl 529 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
31 simpl 109 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
32 op1stg 6145 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3330, 31, 32syl2anc 411 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3429, 33eqtrd 2210 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
35 snidg 3620 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3635adantr 276 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37 opelxpi 4655 . . . . . . . . . . . 12 ((𝑦𝐴𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3830, 36, 37syl2anc 411 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3928, 38eqeltrd 2254 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
4034, 39jca 306 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})))
4127, 40impbida 596 . . . . . . . 8 (𝐵𝑉 → (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4214, 41bitr3id 194 . . . . . . 7 (𝐵𝑉 → ((𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
437, 42bitrid 192 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4443mobidv 2062 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
455, 44mpbiri 168 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4645alrimiv 1874 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47 funcnv2 5272 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4846, 47sylibr 134 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
49 dff1o3 5463 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
503, 48, 49sylanbrc 417 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  Vcvv 2737  {csn 3591  cop 3594   class class class wbr 4000   × cxp 4621  ccnv 4622  cres 4625  Fun wfun 5206   Fn wfn 5207  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  1st c1st 6133  2nd c2nd 6134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136
This theorem is referenced by: (None)
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