Theorem f1oexbi 7909

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 3458	. . . . 5 ⊢ 𝑓 ∈ V
2	1	cnvex 7906	. . . 4 ⊢ ◡𝑓 ∈ V
3		f1ocnv 6819	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
4		f1oeq1 6794	. . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴))
5	4	spcegv 3556	. . . 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 1950	. 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)
8		vex 3458	. . . . 5 ⊢ 𝑔 ∈ V
9	8	cnvex 7906	. . . 4 ⊢ ◡𝑔 ∈ V
10		f1ocnv 6819	. . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ◡𝑔:𝐴–1-1-onto→𝐵)
11		f1oeq1 6794	. . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑔:𝐴–1-1-onto→𝐵))
12	11	spcegv 3556	. . . 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 1950	. 2 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
15	7, 14	impbii 211	1 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)

Syntax hints:   ↔ wb 208  ∃wex 1799   ∈ wcel 2142  Vcvv 3454  ^◡ccnv 5646  –1-1-onto→wf1o 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528

This theorem is referenced by:  rusgrnumwlkg  30177  f1ocnt  32999