Theorem swapf1o 5512

Description: Swap is a bijection over the universe. (Contributed by SF, 23-Feb-2015.) (Revised by Scott Fenton, 17-Apr-2021.)

Assertion

Ref	Expression
swapf1o	⊢ Swap :V–1-1-onto→V

Proof of Theorem swapf1o

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun2 5120	. . . 4 ⊢ (Fun Swap ↔ ∀x∀y∀z((x Swap y ∧ x Swap z) → y = z))
2		opeq 4620	. . . . . . . 8 ⊢ y = ⟨ Proj1 y, Proj2 y⟩
3	2	breq2i 4648	. . . . . . 7 ⊢ (x Swap y ↔ x Swap ⟨ Proj1 y, Proj2 y⟩)
4		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
5	4	proj1ex 4594	. . . . . . . 8 ⊢ Proj1 y ∈ V
6	4	proj2ex 4595	. . . . . . . 8 ⊢ Proj2 y ∈ V
7	5, 6	brswap2 4861	. . . . . . 7 ⊢ (x Swap ⟨ Proj1 y, Proj2 y⟩ ↔ x = ⟨ Proj2 y, Proj1 y⟩)
8	3, 7	bitri 240	. . . . . 6 ⊢ (x Swap y ↔ x = ⟨ Proj2 y, Proj1 y⟩)
9		opeq 4620	. . . . . . . 8 ⊢ z = ⟨ Proj1 z, Proj2 z⟩
10	9	breq2i 4648	. . . . . . 7 ⊢ (x Swap z ↔ x Swap ⟨ Proj1 z, Proj2 z⟩)
11		vex 2863	. . . . . . . . 9 ⊢ z ∈ V
12	11	proj1ex 4594	. . . . . . . 8 ⊢ Proj1 z ∈ V
13	11	proj2ex 4595	. . . . . . . 8 ⊢ Proj2 z ∈ V
14	12, 13	brswap2 4861	. . . . . . 7 ⊢ (x Swap ⟨ Proj1 z, Proj2 z⟩ ↔ x = ⟨ Proj2 z, Proj1 z⟩)
15	10, 14	bitri 240	. . . . . 6 ⊢ (x Swap z ↔ x = ⟨ Proj2 z, Proj1 z⟩)
16		eqtr2 2371	. . . . . . 7 ⊢ ((x = ⟨ Proj2 y, Proj1 y⟩ ∧ x = ⟨ Proj2 z, Proj1 z⟩) → ⟨ Proj2 y, Proj1 y⟩ = ⟨ Proj2 z, Proj1 z⟩)
17		ancom 437	. . . . . . . 8 ⊢ (( Proj2 y = Proj2 z ∧ Proj1 y = Proj1 z) ↔ ( Proj1 y = Proj1 z ∧ Proj2 y = Proj2 z))
18		opth 4603	. . . . . . . 8 ⊢ (⟨ Proj2 y, Proj1 y⟩ = ⟨ Proj2 z, Proj1 z⟩ ↔ ( Proj2 y = Proj2 z ∧ Proj1 y = Proj1 z))
19	2, 9	eqeq12i 2366	. . . . . . . . 9 ⊢ (y = z ↔ ⟨ Proj1 y, Proj2 y⟩ = ⟨ Proj1 z, Proj2 z⟩)
20		opth 4603	. . . . . . . . 9 ⊢ (⟨ Proj1 y, Proj2 y⟩ = ⟨ Proj1 z, Proj2 z⟩ ↔ ( Proj1 y = Proj1 z ∧ Proj2 y = Proj2 z))
21	19, 20	bitri 240	. . . . . . . 8 ⊢ (y = z ↔ ( Proj1 y = Proj1 z ∧ Proj2 y = Proj2 z))
22	17, 18, 21	3bitr4i 268	. . . . . . 7 ⊢ (⟨ Proj2 y, Proj1 y⟩ = ⟨ Proj2 z, Proj1 z⟩ ↔ y = z)
23	16, 22	sylib 188	. . . . . 6 ⊢ ((x = ⟨ Proj2 y, Proj1 y⟩ ∧ x = ⟨ Proj2 z, Proj1 z⟩) → y = z)
24	8, 15, 23	syl2anb 465	. . . . 5 ⊢ ((x Swap y ∧ x Swap z) → y = z)
25	24	gen2 1547	. . . 4 ⊢ ∀y∀z((x Swap y ∧ x Swap z) → y = z)
26	1, 25	mpgbir 1550	. . 3 ⊢ Fun Swap
27		eqv 3566	. . . 4 ⊢ (dom Swap = V ↔ ∀x x ∈ dom Swap )
28		opeq 4620	. . . . 5 ⊢ x = ⟨ Proj1 x, Proj2 x⟩
29		eqid 2353	. . . . . . 7 ⊢ ⟨ Proj1 x, Proj2 x⟩ = ⟨ Proj1 x, Proj2 x⟩
30		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
31	30	proj2ex 4595	. . . . . . . 8 ⊢ Proj2 x ∈ V
32	30	proj1ex 4594	. . . . . . . 8 ⊢ Proj1 x ∈ V
33	31, 32	brswap2 4861	. . . . . . 7 ⊢ (⟨ Proj1 x, Proj2 x⟩ Swap ⟨ Proj2 x, Proj1 x⟩ ↔ ⟨ Proj1 x, Proj2 x⟩ = ⟨ Proj1 x, Proj2 x⟩)
34	29, 33	mpbir 200	. . . . . 6 ⊢ ⟨ Proj1 x, Proj2 x⟩ Swap ⟨ Proj2 x, Proj1 x⟩
35		breldm 4912	. . . . . 6 ⊢ (⟨ Proj1 x, Proj2 x⟩ Swap ⟨ Proj2 x, Proj1 x⟩ → ⟨ Proj1 x, Proj2 x⟩ ∈ dom Swap )
36	34, 35	ax-mp 5	. . . . 5 ⊢ ⟨ Proj1 x, Proj2 x⟩ ∈ dom Swap
37	28, 36	eqeltri 2423	. . . 4 ⊢ x ∈ dom Swap
38	27, 37	mpgbir 1550	. . 3 ⊢ dom Swap = V
39		df-fn 4791	. . 3 ⊢ ( Swap Fn V ↔ (Fun Swap ∧ dom Swap = V))
40	26, 38, 39	mpbir2an 886	. 2 ⊢ Swap Fn V
41		cnvswap 5511	. . . 4 ⊢ ◡ Swap = Swap
42	41	fneq1i 5179	. . 3 ⊢ (◡ Swap Fn V ↔ Swap Fn V)
43	40, 42	mpbir 200	. 2 ⊢ ◡ Swap Fn V
44		dff1o4 5295	. 2 ⊢ ( Swap :V–1-1-onto→V ↔ ( Swap Fn V ∧ ◡ Swap Fn V))
45	40, 43, 44	mpbir2an 886	1 ⊢ Swap :V–1-1-onto→V

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640   Swap cswap 4719  ^◡ccnv 4772  dom cdm 4773  Fun wfun 4776   Fn wfn 4777  –1-1-onto→wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  swapres  5513