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Theorem projf1o 39883
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 4349 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
43adantr 472 . . . . . . 7 ((𝜑𝑦𝐵) → 𝐴 ∈ {𝐴})
5 simpr 479 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
6 opelxpi 5303 . . . . . . 7 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
74, 5, 6syl2anc 696 . . . . . 6 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
8 projf1o.2 . . . . . . 7 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
9 opeq2 4552 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
109cbvmptv 4900 . . . . . . 7 (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
118, 10eqtri 2780 . . . . . 6 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
127, 11fmptd 6546 . . . . 5 (𝜑𝐹:𝐵⟶({𝐴} × 𝐵))
13 simpl1 1228 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝜑)
147elexd 3352 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ V)
1511fvmpt2 6451 . . . . . . . . . . . . . 14 ((𝑦𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ V) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
165, 14, 15syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑦𝐵) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
1716eqcomd 2764 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
18173adant3 1127 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
1918adantr 472 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
20 simpr 479 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑦) = (𝐹𝑧))
2111a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩))
22 opeq2 4552 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
2322adantl 473 . . . . . . . . . . . . 13 (((𝜑𝑧𝐵) ∧ 𝑦 = 𝑧) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
24 simpr 479 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
25 opex 5079 . . . . . . . . . . . . . 14 𝐴, 𝑧⟩ ∈ V
2625a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
2721, 23, 24, 26fvmptd 6448 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
28273adant2 1126 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2928adantr 472 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
3019, 20, 293eqtrd 2796 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
31 simpr 479 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
32 vex 3341 . . . . . . . . . . . . . 14 𝑧 ∈ V
3332a1i 11 . . . . . . . . . . . . 13 (𝜑𝑧 ∈ V)
34 opthg2 5094 . . . . . . . . . . . . 13 ((𝐴𝑉𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
351, 33, 34syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3635adantr 472 . . . . . . . . . . 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 482 . . . . . . . . 9 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
3913, 30, 38syl2anc 696 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝑦 = 𝑧)
4039ex 449 . . . . . . 7 ((𝜑𝑦𝐵𝑧𝐵) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
41403expb 1114 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4241ralrimivva 3107 . . . . 5 (𝜑 → ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4312, 42jca 555 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
44 dff13 6673 . . . 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 479 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → 𝑧 ∈ ({𝐴} × 𝐵))
47 elsnxp 5836 . . . . . . . . . 10 (𝐴𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
481, 47syl 17 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
4948adantr 472 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
5046, 49mpbid 222 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
5116adantr 472 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
52 id 22 . . . . . . . . . . . . 13 (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
5352eqcomd 2764 . . . . . . . . . . . 12 (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
5453adantl 473 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
5551, 54eqtr2d 2793 . . . . . . . . . 10 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹𝑦))
5655ex 449 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5756adantlr 753 . . . . . . . 8 (((𝜑𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5857reximdva 3153 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5950, 58mpd 15 . . . . . 6 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = (𝐹𝑦))
6059ralrimiva 3102 . . . . 5 (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦))
6112, 60jca 555 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
62 dffo3 6535 . . . 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 555 . 2 (𝜑 → (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
65 df-f1o 6054 . 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 1072   = wceq 1630  wcel 2137  wral 3048  wrex 3049  Vcvv 3338  {csn 4319  cop 4325  cmpt 4879   × cxp 5262  wf 6043  1-1wf1 6044  ontowfo 6045  1-1-ontowf1o 6046  cfv 6047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055
This theorem is referenced by:  sge0xp  41147
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