Theorem fntpg 6625

Description: Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Assertion

Ref	Expression
fntpg	⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})

Proof of Theorem fntpg

Step	Hyp	Ref	Expression
1		funtpg 6620	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})
2		dmsnopg 6228	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐹 → dom {⟨𝑋, 𝐴⟩} = {𝑋})
3	2	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) → dom {⟨𝑋, 𝐴⟩} = {𝑋})
4		dmsnopg 6228	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐺 → dom {⟨𝑌, 𝐵⟩} = {𝑌})
5	4	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) → dom {⟨𝑌, 𝐵⟩} = {𝑌})
6	3, 5	jca 521	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) → (dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}))
7	6	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}))
8		uneq12 4165	. . . . . . 7 ⊢ ((dom {⟨𝑋, 𝐴⟩} = {𝑋} ∧ dom {⟨𝑌, 𝐵⟩} = {𝑌}) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = ({𝑋} ∪ {𝑌}))
9	7, 8	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = ({𝑋} ∪ {𝑌}))
10		df-pr 4591	. . . . . 6 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
11	9, 10	eqtr4di 2816	. . . . 5 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
12		df-pr 4591	. . . . . . . 8 ⊢ {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩})
13	12	dmeqi 5894	. . . . . . 7 ⊢ dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩})
14	13	eqeq1i 2768	. . . . . 6 ⊢ (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌} ↔ dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
15		dmun 5900	. . . . . . 7 ⊢ dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩})
16	15	eqeq1i 2768	. . . . . 6 ⊢ (dom ({⟨𝑋, 𝐴⟩} ∪ {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌} ↔ (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
17	14, 16	bitri 278	. . . . 5 ⊢ (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌} ↔ (dom {⟨𝑋, 𝐴⟩} ∪ dom {⟨𝑌, 𝐵⟩}) = {𝑋, 𝑌})
18	11, 17	sylibr 237	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} = {𝑋, 𝑌})
19		dmsnopg 6228	. . . . . 6 ⊢ (𝐶 ∈ 𝐻 → dom {⟨𝑍, 𝐶⟩} = {𝑍})
20	19	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) → dom {⟨𝑍, 𝐶⟩} = {𝑍})
21	20	3ad2ant2 1152	. . . 4 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → dom {⟨𝑍, 𝐶⟩} = {𝑍})
22	18, 21	uneq12d 4171	. . 3 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩}) = ({𝑋, 𝑌} ∪ {𝑍}))
23		df-tp 4593	. . . . 5 ⊢ {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩})
24	23	dmeqi 5894	. . . 4 ⊢ dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = dom ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩})
25		dmun 5900	. . . 4 ⊢ dom ({⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ {⟨𝑍, 𝐶⟩}) = (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩})
26	24, 25	eqtri 2786	. . 3 ⊢ dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = (dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩} ∪ dom {⟨𝑍, 𝐶⟩})
27		df-tp 4593	. . 3 ⊢ {𝑋, 𝑌, 𝑍} = ({𝑋, 𝑌} ∪ {𝑍})
28	22, 26, 27	3eqtr4g 2823	. 2 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → dom {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} = {𝑋, 𝑌, 𝑍})
29		df-fn 6564	. 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}))
30	1, 28, 29	sylanbrc 595	1 ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∪ cun 4152  {csn 4588  {cpr 4590  {ctp 4592  ⟨cop 4838  dom cdm 5656  Fun wfun 6555   Fn wfn 6556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-fun 6563  df-fn 6564

This theorem is used by: (None)