Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  projf1o Structured version   Visualization version   GIF version

Theorem projf1o 40189
Description: A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
projf1o.1 (𝜑𝐴𝑉)
projf1o.2 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
Assertion
Ref Expression
projf1o (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem projf1o
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 projf1o.1 . . . . . . . . 9 (𝜑𝐴𝑉)
2 snidg 4429 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
43adantr 474 . . . . . . 7 ((𝜑𝑦𝐵) → 𝐴 ∈ {𝐴})
5 simpr 479 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
6 opelxpi 5383 . . . . . . 7 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
74, 5, 6syl2anc 579 . . . . . 6 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
8 projf1o.2 . . . . . . 7 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
9 opeq2 4626 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
109cbvmptv 4975 . . . . . . 7 (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
118, 10eqtri 2849 . . . . . 6 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
127, 11fmptd 6638 . . . . 5 (𝜑𝐹:𝐵⟶({𝐴} × 𝐵))
13 simpl1 1246 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝜑)
148, 9, 5, 7fvmptd3 6555 . . . . . . . . . . . . 13 ((𝜑𝑦𝐵) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
1514eqcomd 2831 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
16153adant3 1166 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
1716adantr 474 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
18 simpr 479 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑦) = (𝐹𝑧))
19 opeq2 4626 . . . . . . . . . . . . 13 (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
20 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
21 opex 5155 . . . . . . . . . . . . . 14 𝐴, 𝑧⟩ ∈ V
2221a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
2311, 19, 20, 22fvmptd3 6555 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
24233adant2 1165 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2524adantr 474 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2617, 18, 253eqtrd 2865 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
27 simpr 479 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
28 vex 3417 . . . . . . . . . . . . . 14 𝑧 ∈ V
2928a1i 11 . . . . . . . . . . . . 13 (𝜑𝑧 ∈ V)
30 opthg2 5170 . . . . . . . . . . . . 13 ((𝐴𝑉𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
311, 29, 30syl2anc 579 . . . . . . . . . . . 12 (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3231adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3327, 32mpbid 224 . . . . . . . . . 10 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (𝐴 = 𝐴𝑦 = 𝑧))
3433simprd 491 . . . . . . . . 9 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
3513, 26, 34syl2anc 579 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝑦 = 𝑧)
3635ex 403 . . . . . . 7 ((𝜑𝑦𝐵𝑧𝐵) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
37363expb 1153 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3837ralrimivva 3180 . . . . 5 (𝜑 → ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3912, 38jca 507 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
40 dff13 6772 . . . 4 (𝐹:𝐵1-1→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
4139, 40sylibr 226 . . 3 (𝜑𝐹:𝐵1-1→({𝐴} × 𝐵))
42 simpr 479 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → 𝑧 ∈ ({𝐴} × 𝐵))
43 elsnxp 5922 . . . . . . . . . 10 (𝐴𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
441, 43syl 17 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
4544adantr 474 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
4642, 45mpbid 224 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
4714adantr 474 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
48 id 22 . . . . . . . . . . . . 13 (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
4948eqcomd 2831 . . . . . . . . . . . 12 (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
5049adantl 475 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
5147, 50eqtr2d 2862 . . . . . . . . . 10 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹𝑦))
5251ex 403 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5352adantlr 706 . . . . . . . 8 (((𝜑𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5453reximdva 3225 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5546, 54mpd 15 . . . . . 6 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = (𝐹𝑦))
5655ralrimiva 3175 . . . . 5 (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦))
5712, 56jca 507 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
58 dffo3 6628 . . . 4 (𝐹:𝐵onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5957, 58sylibr 226 . . 3 (𝜑𝐹:𝐵onto→({𝐴} × 𝐵))
6041, 59jca 507 . 2 (𝜑 → (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
61 df-f1o 6134 . 2 (𝐹:𝐵1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
6260, 61sylibr 226 1 (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  Vcvv 3414  {csn 4399  cop 4405  cmpt 4954   × cxp 5344  wf 6123  1-1wf1 6124  ontowfo 6125  1-1-ontowf1o 6126  cfv 6127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135
This theorem is referenced by:  sge0xp  41431
  Copyright terms: Public domain W3C validator