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Theorem projf1o 41448
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 . . . . . . 7 (𝜑𝐴𝑉)
2 snidg 4591 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
31, 2syl 17 . . . . . 6 (𝜑𝐴 ∈ {𝐴})
43adantr 483 . . . . 5 ((𝜑𝑦𝐵) → 𝐴 ∈ {𝐴})
5 simpr 487 . . . . 5 ((𝜑𝑦𝐵) → 𝑦𝐵)
64, 5opelxpd 5586 . . . 4 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
7 projf1o.2 . . . . 5 𝐹 = (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩)
8 opeq2 4796 . . . . . 6 (𝑥 = 𝑦 → ⟨𝐴, 𝑥⟩ = ⟨𝐴, 𝑦⟩)
98cbvmptv 5160 . . . . 5 (𝑥𝐵 ↦ ⟨𝐴, 𝑥⟩) = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
107, 9eqtri 2842 . . . 4 𝐹 = (𝑦𝐵 ↦ ⟨𝐴, 𝑦⟩)
116, 10fmptd 6871 . . 3 (𝜑𝐹:𝐵⟶({𝐴} × 𝐵))
12 simpl1 1186 . . . . . . 7 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝜑)
137, 8, 5, 6fvmptd3 6784 . . . . . . . . . . 11 ((𝜑𝑦𝐵) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
1413eqcomd 2825 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
15143adant3 1127 . . . . . . . . 9 ((𝜑𝑦𝐵𝑧𝐵) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
1615adantr 483 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = (𝐹𝑦))
17 simpr 487 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑦) = (𝐹𝑧))
18 opeq2 4796 . . . . . . . . . . 11 (𝑦 = 𝑧 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
19 simpr 487 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → 𝑧𝐵)
20 opex 5347 . . . . . . . . . . . 12 𝐴, 𝑧⟩ ∈ V
2120a1i 11 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → ⟨𝐴, 𝑧⟩ ∈ V)
2210, 18, 19, 21fvmptd3 6784 . . . . . . . . . 10 ((𝜑𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
23223adant2 1126 . . . . . . . . 9 ((𝜑𝑦𝐵𝑧𝐵) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2423adantr 483 . . . . . . . 8 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → (𝐹𝑧) = ⟨𝐴, 𝑧⟩)
2516, 17, 243eqtrd 2858 . . . . . . 7 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩)
26 vex 3496 . . . . . . . . . 10 𝑧 ∈ V
2726a1i 11 . . . . . . . . 9 (𝜑𝑧 ∈ V)
28 opthg2 5362 . . . . . . . . 9 ((𝐴𝑉𝑧 ∈ V) → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
291, 27, 28syl2anc 586 . . . . . . . 8 (𝜑 → (⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩ ↔ (𝐴 = 𝐴𝑦 = 𝑧)))
3029simplbda 502 . . . . . . 7 ((𝜑 ∧ ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝑧⟩) → 𝑦 = 𝑧)
3112, 25, 30syl2anc 586 . . . . . 6 (((𝜑𝑦𝐵𝑧𝐵) ∧ (𝐹𝑦) = (𝐹𝑧)) → 𝑦 = 𝑧)
3231ex 415 . . . . 5 ((𝜑𝑦𝐵𝑧𝐵) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
33323expb 1115 . . . 4 ((𝜑 ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
3433ralrimivva 3189 . . 3 (𝜑 → ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
35 dff13 7005 . . 3 (𝐹:𝐵1-1→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑦𝐵𝑧𝐵 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
3611, 34, 35sylanbrc 585 . 2 (𝜑𝐹:𝐵1-1→({𝐴} × 𝐵))
37 elsnxp 6135 . . . . . . 7 (𝐴𝑉 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
381, 37syl 17 . . . . . 6 (𝜑 → (𝑧 ∈ ({𝐴} × 𝐵) ↔ ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩))
3938biimpa 479 . . . . 5 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩)
4013adantr 483 . . . . . . . . 9 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → (𝐹𝑦) = ⟨𝐴, 𝑦⟩)
41 id 22 . . . . . . . . . . 11 (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = ⟨𝐴, 𝑦⟩)
4241eqcomd 2825 . . . . . . . . . 10 (𝑧 = ⟨𝐴, 𝑦⟩ → ⟨𝐴, 𝑦⟩ = 𝑧)
4342adantl 484 . . . . . . . . 9 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → ⟨𝐴, 𝑦⟩ = 𝑧)
4440, 43eqtr2d 2855 . . . . . . . 8 (((𝜑𝑦𝐵) ∧ 𝑧 = ⟨𝐴, 𝑦⟩) → 𝑧 = (𝐹𝑦))
4544ex 415 . . . . . . 7 ((𝜑𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
4645adantlr 713 . . . . . 6 (((𝜑𝑧 ∈ ({𝐴} × 𝐵)) ∧ 𝑦𝐵) → (𝑧 = ⟨𝐴, 𝑦⟩ → 𝑧 = (𝐹𝑦)))
4746reximdva 3272 . . . . 5 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → (∃𝑦𝐵 𝑧 = ⟨𝐴, 𝑦⟩ → ∃𝑦𝐵 𝑧 = (𝐹𝑦)))
4839, 47mpd 15 . . . 4 ((𝜑𝑧 ∈ ({𝐴} × 𝐵)) → ∃𝑦𝐵 𝑧 = (𝐹𝑦))
4948ralrimiva 3180 . . 3 (𝜑 → ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦))
50 dffo3 6861 . . 3 (𝐹:𝐵onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵⟶({𝐴} × 𝐵) ∧ ∀𝑧 ∈ ({𝐴} × 𝐵)∃𝑦𝐵 𝑧 = (𝐹𝑦)))
5111, 49, 50sylanbrc 585 . 2 (𝜑𝐹:𝐵onto→({𝐴} × 𝐵))
52 df-f1o 6355 . 2 (𝐹:𝐵1-1-onto→({𝐴} × 𝐵) ↔ (𝐹:𝐵1-1→({𝐴} × 𝐵) ∧ 𝐹:𝐵onto→({𝐴} × 𝐵)))
5336, 51, 52sylanbrc 585 1 (𝜑𝐹:𝐵1-1-onto→({𝐴} × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  wrex 3137  Vcvv 3493  {csn 4559  cop 4565  cmpt 5137   × cxp 5546  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  sge0xp  42701
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