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Theorem tglnfn 28302
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 6898 . . . . . . 7 𝑃 ∈ V
32rabex 5325 . . . . . 6 {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
43rgen2w 3060 . . . . 5 βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ (𝑃 βˆ– {π‘₯}){𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))} ∈ V
5 eqid 2726 . . . . . 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 8049 . . . . 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 6710 . . . 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 6160 . . . 4 βˆͺ π‘₯ ∈ 𝑃 ({π‘₯} Γ— (𝑃 βˆ– {π‘₯})) = ((𝑃 Γ— 𝑃) βˆ– I )
1110fneq2i 6640 . . 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 28301 . . 3 (𝐺 ∈ TarskiG β†’ 𝐿 = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
1615fneq1d 6635 . 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 258 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 3055  {crab 3426  Vcvv 3468   βˆ– cdif 3940  {csn 4623  βˆͺ ciun 4990   I cid 5566   Γ— cxp 5667   Fn wfn 6531  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  Basecbs 17151  TarskiGcstrkg 28182  Itvcitv 28188  LineGclng 28189
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-trkg 28208
This theorem is referenced by:  tglngne  28305  tgelrnln  28385
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