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Theorem projf1o 38164
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 4152 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2syl 17 . . . . . . . 8 (𝜑𝐴 ∈ {𝐴})
43adantr 479 . . . . . . 7 ((𝜑𝑦𝐵) → 𝐴 ∈ {𝐴})
5 simpr 475 . . . . . . 7 ((𝜑𝑦𝐵) → 𝑦𝐵)
6 opelxpi 5061 . . . . . . 7 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
74, 5, 6syl2anc 690 . . . . . 6 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
8 projf1o.2 . . . . . . 7 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
9 opeq2 4335 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
109cbvmptv 4672 . . . . . . 7 (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
118, 10eqtri 2631 . . . . . 6 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
127, 11fmptd 6276 . . . . 5 (𝜑𝐹:𝐵⟶({𝐴} × 𝐵))
13 simpl1 1056 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝜑)
147elexd 3186 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ V)
1511fvmpt2 6184 . . . . . . . . . . . . . 14 ((𝑦𝐵 ∧ ⟨𝐴, 𝑦⟩ ∈ V) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
165, 14, 15syl2anc 690 . . . . . . . . . . . . 13 ((𝜑𝑦𝐵) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
1716eqcomd 2615 . . . . . . . . . . . 12 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
18173adant3 1073 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
1918adantr 479 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
20 simpr 475 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑦) = (𝐹𝑧))
2111a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩))
22 opeq2 4335 . . . . . . . . . . . . . 14 (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
2322adantl 480 . . . . . . . . . . . . 13 (((𝜑𝑧𝐵) ∧ 𝑦 = 𝑧) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
24 simpr 475 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
25 opex 4852 . . . . . . . . . . . . . 14 𝐴, 𝑧⟩ ∈ V
2625a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
2721, 23, 24, 26fvmptd 6181 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
28273adant2 1072 . . . . . . . . . . 11 ((𝜑𝑦𝐵𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2928adantr 479 . . . . . . . . . 10 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
3019, 20, 293eqtrd 2647 . . . . . . . . 9 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
31 simpr 475 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
32 vex 3175 . . . . . . . . . . . . . 14 𝑧 ∈ V
3332a1i 11 . . . . . . . . . . . . 13 (𝜑𝑧 ∈ V)
34 opthg2 4867 . . . . . . . . . . . . 13 ((𝐴𝑉𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
351, 33, 34syl2anc 690 . . . . . . . . . . . 12 (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3635adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3731, 36mpbid 220 . . . . . . . . . 10 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → (𝐴 = 𝐴𝑦 = 𝑧))
3837simprd 477 . . . . . . . . 9 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
3913, 30, 38syl2anc 690 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝑦 = 𝑧)
4039ex 448 . . . . . . 7 ((𝜑𝑦𝐵𝑧𝐵) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
41403expb 1257 . . . . . 6 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4241ralrimivva 2953 . . . . 5 (𝜑 → ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
4312, 42jca 552 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
44 dff13 6393 . . . 4 (𝐹:𝐵1-1→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
4543, 44sylibr 222 . . 3 (𝜑𝐹:𝐵1-1→({𝐴} × 𝐵))
46 simpr 475 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → 𝑧 ∈ ({𝐴} × 𝐵))
47 elsnxp 5579 . . . . . . . . . 10 (𝐴𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
481, 47syl 17 . . . . . . . . 9 (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
4948adantr 479 . . . . . . . 8 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
5046, 49mpbid 220 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
5116adantr 479 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
52 id 22 . . . . . . . . . . . . 13 (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
5352eqcomd 2615 . . . . . . . . . . . 12 (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
5453adantl 480 . . . . . . . . . . 11 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
5551, 54eqtr2d 2644 . . . . . . . . . 10 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹𝑦))
5655ex 448 . . . . . . . . 9 ((𝜑𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5756adantlr 746 . . . . . . . 8 (((𝜑𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
5857reximdva 2999 . . . . . . 7 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5950, 58mpd 15 . . . . . 6 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = (𝐹𝑦))
6059ralrimiva 2948 . . . . 5 (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦))
6112, 60jca 552 . . . 4 (𝜑 → (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
62 dffo3 6266 . . . 4 (𝐹:𝐵onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
6361, 62sylibr 222 . . 3 (𝜑𝐹:𝐵onto→({𝐴} × 𝐵))
6445, 63jca 552 . 2 (𝜑 → (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
65 df-f1o 5796 . 2 (𝐹:𝐵1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
6664, 65sylibr 222 1 (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  {csn 4124  cop 4130  cmpt 4637   × cxp 5025  wf 5785  1-1wf1 5786  ontowfo 5787  1-1-ontowf1o 5788  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797
This theorem is referenced by:  sge0xp  39105
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