Theorem f1oprswap 6890

Description: A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)

Assertion

Ref	Expression
f1oprswap	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵})

Proof of Theorem f1oprswap

Step	Hyp	Ref	Expression
1		f1osng 6887	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {⟨𝐴, 𝐴⟩}:{𝐴}⟶1-1onto{𝐴})
2	1	anidms 577	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {⟨𝐴, 𝐴⟩}:{𝐴}⟶1-1onto{𝐴})
3	2	ad2antrr 739	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐴⟩}:{𝐴}⟶1-1onto{𝐴})
4		dfsn2 4597	. . . . . 6 ⊢ {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩}
5		opeq2 4844	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐴, 𝐵⟩)
6		opeq1 4843	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
7	5, 6	preq12d 4702	. . . . . 6 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩, ⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
8	4, 7	eqtrid 2810	. . . . 5 ⊢ (𝐴 = 𝐵 → {⟨𝐴, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
9		dfsn2 4597	. . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴}
10		preq2 4695	. . . . . 6 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
11	9, 10	eqtrid 2810	. . . . 5 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐴, 𝐵})
12	8, 11, 11	f1oeq123d 6837	. . . 4 ⊢ (𝐴 = 𝐵 → ({⟨𝐴, 𝐴⟩}:{𝐴}⟶1-1onto{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵}))
13	12	adantl 487	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → ({⟨𝐴, 𝐴⟩}:{𝐴}⟶1-1onto{𝐴} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵}))
14	3, 13	mpbid 235	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵})
15		simpll 779	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑉)
16		simplr 781	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑊)
17		simpr 490	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
18		fnprg 6618	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
19	15, 16, 16, 15, 17, 18	syl221anc 1409	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
20		cnvsng 6236	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21		cnvsng 6236	. . . . . . . . . 10 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ◡{⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
22	21	ancoms 464	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
23	20, 22	uneq12d 4171	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐵, 𝐴⟩}) = ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}))
24		uncom 4160	. . . . . . . 8 ⊢ ({⟨𝐵, 𝐴⟩} ∪ {⟨𝐴, 𝐵⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
25	23, 24	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
26	25	adantr 486	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐵, 𝐴⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}))
27		df-pr 4591	. . . . . . . 8 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
28	27	cnveqi 5858	. . . . . . 7 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩})
29		cnvun 6144	. . . . . . 7 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐵, 𝐴⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐵, 𝐴⟩})
30	28, 29	eqtri 2786	. . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐵, 𝐴⟩})
31	26, 30, 27	3eqtr4g 2823	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩})
32	31	fneq1d 6651	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ↔ {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
33	19, 32	mpbird 260	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵})
34		dff1o4 6852	. . 3 ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵} ↔ ({⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵} ∧ ◡{⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩} Fn {𝐴, 𝐵}))
35	19, 33, 34	sylanbrc 595	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵})
36	14, 35	pm2.61dane 3046	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}⟶1-1onto{𝐴, 𝐵})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∪ cun 4152  {csn 4588  {cpr 4590  ⟨cop 4838  ^◡ccnv 5654   Fn wfn 6550  ⟶1-1ontowf1o 6554

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 5252  ax-pr 5395

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-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562

This theorem is used by:  fveqf1o  7330  f1ofvswap  7334  symg2bas  19634  subfacp1lem2a  36111