Theorem f1oexbi 7915

Description: There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.)

Assertion

Ref	Expression
f1oexbi	⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)

Distinct variable groups:   𝐴,𝑓,𝑔   𝐵,𝑓,𝑔

Proof of Theorem f1oexbi

Step	Hyp	Ref	Expression
1		vex 3472	. . . . 5 ⊢ 𝑓 ∈ V
2	1	cnvex 7912	. . . 4 ⊢ ◡𝑓 ∈ V
3		f1ocnv 6838	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
4		f1oeq1 6814	. . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
5	4	spcegv 3581	. . . 4 ⊢ (◡𝑓 ∈ V → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴))
6	2, 3, 5	mpsyl 68	. . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
7	6	exlimiv 1925	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
8		vex 3472	. . . . 5 ⊢ 𝑔 ∈ V
9	8	cnvex 7912	. . . 4 ⊢ ◡𝑔 ∈ V
10		f1ocnv 6838	. . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ◡𝑔:𝐴–1-1-onto→𝐵)
11		f1oeq1 6814	. . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑔:𝐴–1-1-onto→𝐵))
12	11	spcegv 3581	. . . 4 ⊢ (◡𝑔 ∈ V → (◡𝑔:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
13	9, 10, 12	mpsyl 68	. . 3 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
14	13	exlimiv 1925	. 2 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
15	7, 14	impbii 208	1 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)

Syntax hints:   ↔ wb 205  ∃wex 1773   ∈ wcel 2098  Vcvv 3468  ^◡ccnv 5668  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is referenced by:  rusgrnumwlkg  29735  f1ocnt  32517