Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fntpg Structured version   Visualization version   GIF version

Theorem fntpg 6385
 Description: Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
Assertion
Ref Expression
fntpg (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})

Proof of Theorem fntpg
StepHypRef Expression
1 funtpg 6380 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})
2 dmsnopg 6038 . . . . . . . . . 10 (𝐴𝐹 → dom {⟨𝑋, 𝐴⟩} = {𝑋})
323ad2ant1 1130 . . . . . . . . 9 ((𝐴𝐹𝐵𝐺𝐶𝐻) → dom {⟨𝑋, 𝐴⟩} = {𝑋})
4 dmsnopg 6038 . . . . . . . . . 10 (𝐵𝐺 → dom {⟨𝑌, 𝐵⟩} = {𝑌})
543ad2ant2 1131 . . . . . . . . 9 ((𝐴𝐹𝐵𝐺𝐶𝐻) → dom {⟨𝑌, 𝐵⟩} = {𝑌})
63, 5jca 515 . . . . . . . 8 ((𝐴𝐹𝐵𝐺𝐶𝐻) → (dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}))
763ad2ant2 1131 . . . . . . 7 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}))
8 uneq12 4085 . . . . . . 7 ((dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = ({𝑋} ∪ {𝑌}))
97, 8syl 17 . . . . . 6 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = ({𝑋} ∪ {𝑌}))
10 df-pr 4528 . . . . . 6 {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
119, 10eqtr4di 2851 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
12 df-pr 4528 . . . . . . . 8 {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩})
1312dmeqi 5738 . . . . . . 7 dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩})
1413eqeq1i 2803 . . . . . 6 (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌} ↔ dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
15 dmun 5744 . . . . . . 7 dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩})
1615eqeq1i 2803 . . . . . 6 (dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌} ↔ (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
1714, 16bitri 278 . . . . 5 (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌} ↔ (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
1811, 17sylibr 237 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌})
19 dmsnopg 6038 . . . . . 6 (𝐶𝐻 → dom {⟨𝑍, 𝐶⟩} = {𝑍})
20193ad2ant3 1132 . . . . 5 ((𝐴𝐹𝐵𝐺𝐶𝐻) → dom {⟨𝑍, 𝐶⟩} = {𝑍})
21203ad2ant2 1131 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → dom {⟨𝑍, 𝐶⟩} = {𝑍})
2218, 21uneq12d 4091 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩}) = ({𝑋, 𝑌} ∪ {𝑍}))
23 df-tp 4530 . . . . 5 {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩})
2423dmeqi 5738 . . . 4 dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = dom ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩})
25 dmun 5744 . . . 4 dom ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩}) = (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩})
2624, 25eqtri 2821 . . 3 dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩})
27 df-tp 4530 . . 3 {𝑋, 𝑌, 𝑍} = ({𝑋, 𝑌} ∪ {𝑍})
2822, 26, 273eqtr4g 2858 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = {𝑋, 𝑌, 𝑍})
29 df-fn 6328 . 2 ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍} ↔ (Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} ∧ dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = {𝑋, 𝑌, 𝑍}))
301, 28, 29sylanbrc 586 1 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴𝐹𝐵𝐺𝐶𝐻) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∪ cun 3879  {csn 4525  {cpr 4527  {ctp 4529  ⟨cop 4531  dom cdm 5520  Fun wfun 6319   Fn wfn 6320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-fun 6327  df-fn 6328 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator