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Theorem tglnfn 28369
Description: Lines as functions. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglng.p 𝑃 = (Baseβ€˜πΊ)
tglng.l 𝐿 = (LineGβ€˜πΊ)
tglng.i 𝐼 = (Itvβ€˜πΊ)
Assertion
Ref Expression
tglnfn (𝐺 ∈ TarskiG β†’ 𝐿 Fn ((𝑃 Γ— 𝑃) βˆ– I ))

Proof of Theorem tglnfn
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglng.p . . . . . . . 8 𝑃 = (Baseβ€˜πΊ)
21fvexi 6914 . . . . . . 7 𝑃 ∈ V
32rabex 5336 . . . . . 6 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
43rgen2w 3062 . . . . 5 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ (𝑃 βˆ– {π‘₯}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
5 eqid 2727 . . . . . 6 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
65fmpox 8075 . . . . 5 (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ (𝑃 βˆ– {π‘₯}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V ↔ (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}):βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯}))⟢V)
74, 6mpbi 229 . . . 4 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}):βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯}))⟢V
8 ffn 6725 . . . 4 ((π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}):βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯}))⟢V β†’ (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯})))
97, 8ax-mp 5 . . 3 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯}))
10 xpdifid 6175 . . . 4 βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯})) = ((𝑃 Γ— 𝑃) βˆ– I )
1110fneq2i 6655 . . 3 ((π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯})) ↔ (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn ((𝑃 Γ— 𝑃) βˆ– I ))
129, 11mpbi 229 . 2 (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn ((𝑃 Γ— 𝑃) βˆ– I )
13 tglng.l . . . 4 𝐿 = (LineGβ€˜πΊ)
14 tglng.i . . . 4 𝐼 = (Itvβ€˜πΊ)
151, 13, 14tglng 28368 . . 3 (𝐺 ∈ TarskiG β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
1615fneq1d 6650 . 2 (𝐺 ∈ TarskiG β†’ (𝐿 Fn ((𝑃 Γ— 𝑃) βˆ– I ) ↔ (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}) Fn ((𝑃 Γ— 𝑃) βˆ– I )))
1712, 16mpbiri 257 1 (𝐺 ∈ TarskiG β†’ 𝐿 Fn ((𝑃 Γ— 𝑃) βˆ– I ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  {crab 3428  Vcvv 3471   βˆ– cdif 3944  {csn 4630  βˆͺ ciun 4998   I cid 5577   Γ— cxp 5678   Fn wfn 6546  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424   ∈ cmpo 7426  Basecbs 17185  TarskiGcstrkg 28249  Itvcitv 28255  LineGclng 28256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-trkg 28275
This theorem is referenced by:  tglngne  28372  tgelrnln  28452
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