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Theorem 1stconst 7217
 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 snnzg 4283 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 7144 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3368 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2524 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3192 . . . . . . . 8 𝑦 ∈ V
76brres 5367 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
8 fo1st 7140 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6079 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3192 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6198 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 707 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi1i 730 . . . . . . . 8 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})))
15 elxp7 7153 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2686 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpa 501 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (1st𝑥) ∈ 𝐴) → 𝑦𝐴)
1817adantrr 752 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) → 𝑦𝐴)
1918adantrl 751 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑦𝐴)
20 elsni 4170 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 7154 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221an12s 842 . . . . . . . . . . . . . 14 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanr2 684 . . . . . . . . . . . . 13 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵})) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantrrl 759 . . . . . . . . . . . 12 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 554 . . . . . . . . . . 11 (((1st𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylan2b 492 . . . . . . . . . 10 (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 482 . . . . . . . . 9 ((𝐵𝑉 ∧ ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵}))) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 795 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2928fveq2d 6157 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
30 simprl 793 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
31 simpl 473 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
32 op1stg 7132 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3330, 31, 32syl2anc 692 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3429, 33eqtrd 2655 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
35 snidg 4182 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3635adantr 481 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
37 opelxpi 5113 . . . . . . . . . . . 12 ((𝑦𝐴𝐵 ∈ {𝐵}) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3830, 36, 37syl2anc 692 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3928, 38eqeltrd 2698 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
4034, 39jca 554 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})))
4127, 40impbida 876 . . . . . . . 8 (𝐵𝑉 → (((1st𝑥) = 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4214, 41syl5bbr 274 . . . . . . 7 (𝐵𝑉 → ((𝑥1st 𝑦𝑥 ∈ (𝐴 × {𝐵})) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
437, 42syl5bb 272 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4443mobidv 2490 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
455, 44mpbiri 248 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4645alrimiv 1852 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
47 funcnv2 5920 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4846, 47sylibr 224 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
49 dff1o3 6105 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
503, 48, 49sylanbrc 697 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480   ∈ wcel 1987  ∃*wmo 2470   ≠ wne 2790  Vcvv 3189  ∅c0 3896  {csn 4153  ⟨cop 4159   class class class wbr 4618   × cxp 5077  ◡ccnv 5078   ↾ cres 5081  Fun wfun 5846   Fn wfn 5847  –onto→wfo 5850  –1-1-onto→wf1o 5851  ‘cfv 5852  1st c1st 7118  2nd c2nd 7119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1st 7120  df-2nd 7121 This theorem is referenced by:  curry2  7224  domss2  8071  fv1stcnv  31417
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