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Theorem projf1o 39700
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 4239 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
43adantr 480 . . . . . . 7 ((𝜑𝑦𝐵) → 𝐴 ∈ {𝐴})
5 simpr 476 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
6 opelxpi 5182 . . . . . . 7 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
74, 5, 6syl2anc 694 . . . . . 6 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
8 projf1o.2 . . . . . . 7 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
9 opeq2 4434 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
109cbvmptv 4783 . . . . . . 7 (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
118, 10eqtri 2673 . . . . . 6 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
127, 11fmptd 6425 . . . . 5 (𝜑𝐹:𝐵⟶({𝐴} × 𝐵))
13 simpl1 1084 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝜑)
147elexd 3245 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ V)
1511fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑦𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ V) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
165, 14, 15syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑦𝐵) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
1716eqcomd 2657 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
18173adant3 1101 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
1918adantr 480 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
20 simpr 476 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑦) = (𝐹𝑧))
2111a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩))
22 opeq2 4434 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
2322adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑧𝐵) ∧ 𝑦 = 𝑧) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
24 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
25 opex 4962 . . . . . . . . . . . . . 14 𝐴, 𝑧⟩ ∈ V
2625a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
2721, 23, 24, 26fvmptd 6327 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
28273adant2 1100 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2928adantr 480 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
3019, 20, 293eqtrd 2689 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
31 simpr 476 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
32 vex 3234 . . . . . . . . . . . . . 14 𝑧 ∈ V
3332a1i 11 . . . . . . . . . . . . 13 (𝜑𝑧 ∈ V)
34 opthg2 4977 . . . . . . . . . . . . 13 ((𝐴𝑉𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
351, 33, 34syl2anc 694 . . . . . . . . . . . 12 (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3635adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3731, 36mpbid 222 . . . . . . . . . 10 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (𝐴 = 𝐴𝑦 = 𝑧))
3837simprd 478 . . . . . . . . 9 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
3913, 30, 38syl2anc 694 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝑦 = 𝑧)
4039ex 449 . . . . . . 7 ((𝜑𝑦𝐵𝑧𝐵) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
41403expb 1285 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4241ralrimivva 3000 . . . . 5 (𝜑 → ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4312, 42jca 553 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
44 dff13 6552 . . . 4 (𝐹:𝐵1-1→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
4543, 44sylibr 224 . . 3 (𝜑𝐹:𝐵1-1→({𝐴} × 𝐵))
46 simpr 476 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → 𝑧 ∈ ({𝐴} × 𝐵))
47 elsnxp 5715 . . . . . . . . . 10 (𝐴𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
481, 47syl 17 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
4948adantr 480 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
5046, 49mpbid 222 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
5116adantr 480 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
52 id 22 . . . . . . . . . . . . 13 (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
5352eqcomd 2657 . . . . . . . . . . . 12 (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
5453adantl 481 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
5551, 54eqtr2d 2686 . . . . . . . . . 10 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹𝑦))
5655ex 449 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5756adantlr 751 . . . . . . . 8 (((𝜑𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5857reximdva 3046 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5950, 58mpd 15 . . . . . 6 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = (𝐹𝑦))
6059ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦))
6112, 60jca 553 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
62 dffo3 6414 . . . 4 (𝐹:𝐵onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
6361, 62sylibr 224 . . 3 (𝜑𝐹:𝐵onto→({𝐴} × 𝐵))
6445, 63jca 553 . 2 (𝜑 → (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
65 df-f1o 5933 . 2 (𝐹:𝐵1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
6664, 65sylibr 224 1 (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  {csn 4210  cop 4216  cmpt 4762   × cxp 5141  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  sge0xp  40964
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