Theorem funcnvtp 6580

Description: The converse triple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)

Assertion

Ref	Expression
funcnvtp	⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})

Proof of Theorem funcnvtp

Step	Hyp	Ref	Expression
1		simp1 1148	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐴 ∈ 𝑈)
2		simp2 1149	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐶 ∈ 𝑉)
3		simp1 1148	. . . 4 ⊢ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹) → 𝐵 ≠ 𝐷)
4		funcnvpr 6579	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
5	1, 2, 3, 4	syl2an3an 1440	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6		funcnvsn 6567	. . . 4 ⊢ Fun ◡{⟨𝐸, 𝐹⟩}
7	6	a1i 11	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐸, 𝐹⟩})
8		df-rn 5656	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9		rnpropg 6205	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
10	8, 9	eqtr3id 2810	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11	10	3adant3 1144	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12		df-rn 5656	. . . . . . 7 ⊢ ran {⟨𝐸, 𝐹⟩} = dom ◡{⟨𝐸, 𝐹⟩}
13		rnsnopg 6204	. . . . . . 7 ⊢ (𝐸 ∈ 𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
14	12, 13	eqtr3id 2810	. . . . . 6 ⊢ (𝐸 ∈ 𝑊 → dom ◡{⟨𝐸, 𝐹⟩} = {𝐹})
15	14	3ad2ant3 1147	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → dom ◡{⟨𝐸, 𝐹⟩} = {𝐹})
16	11, 15	ineq12d 4173	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom ◡{⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17		disjprsn 4672	. . . . 5 ⊢ ((𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18	17	3adant1 1142	. . . 4 ⊢ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
19	16, 18	sylan9eq 2816	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → (dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom ◡{⟨𝐸, 𝐹⟩}) = ∅)
20		funun 6563	. . 3 ⊢ (((Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∧ Fun ◡{⟨𝐸, 𝐹⟩}) ∧ (dom ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom ◡{⟨𝐸, 𝐹⟩}) = ∅) → Fun (◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ ◡{⟨𝐸, 𝐹⟩}))
21	5, 7, 19, 20	syl21anc 848	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun (◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ ◡{⟨𝐸, 𝐹⟩}))
22		df-tp 4586	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
23	22	cnveqi 5844	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ◡({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24		cnvun 6123	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = (◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ ◡{⟨𝐸, 𝐹⟩})
25	23, 24	eqtri 2784	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = (◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ ◡{⟨𝐸, 𝐹⟩})
26	25	funeqi 6538	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ Fun (◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ ◡{⟨𝐸, 𝐹⟩}))
27	21, 26	sylibr 236	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4581  {cpr 4583  {ctp 4585  ⟨cop 4587  ^◡ccnv 5644  dom cdm 5645  ran crn 5646  Fun wfun 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-fun 6519

This theorem is referenced by:  funcnvs3  14924