Theorem f1oexbi 7974

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⟶1-1onto𝐵 ↔ ∃𝑔 𝑔:𝐵⟶1-1onto𝐴)

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 7971	. . . 4 ⊢ ◡𝑓 ∈ V
3		f1ocnv 6867	. . . 4 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓:𝐵⟶1-1onto𝐴)
4		f1oeq1 6842	. . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵⟶1-1onto𝐴 ↔ ◡𝑓:𝐵⟶1-1onto𝐴))
5	4	spcegv 3555	. . . 4 ⊢ (◡𝑓 ∈ V → (◡𝑓:𝐵⟶1-1onto𝐴 → ∃𝑔 𝑔:𝐵⟶1-1onto𝐴))
6	2, 3, 5	mpsyl 69	. . 3 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ∃𝑔 𝑔:𝐵⟶1-1onto𝐴)
7	6	exlimiv 1965	. 2 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto𝐵 → ∃𝑔 𝑔:𝐵⟶1-1onto𝐴)
8		vex 3458	. . . . 5 ⊢ 𝑔 ∈ V
9	8	cnvex 7971	. . . 4 ⊢ ◡𝑔 ∈ V
10		f1ocnv 6867	. . . 4 ⊢ (𝑔:𝐵⟶1-1onto𝐴 → ◡𝑔:𝐴⟶1-1onto𝐵)
11		f1oeq1 6842	. . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴⟶1-1onto𝐵 ↔ ◡𝑔:𝐴⟶1-1onto𝐵))
12	11	spcegv 3555	. . . 4 ⊢ (◡𝑔 ∈ V → (◡𝑔:𝐴⟶1-1onto𝐵 → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵))
13	9, 10, 12	mpsyl 69	. . 3 ⊢ (𝑔:𝐵⟶1-1onto𝐴 → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
14	13	exlimiv 1965	. 2 ⊢ (∃𝑔 𝑔:𝐵⟶1-1onto𝐴 → ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
15	7, 14	impbii 212	1 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto𝐵 ↔ ∃𝑔 𝑔:𝐵⟶1-1onto𝐴)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  ^◡ccnv 5657  ⟶1-1ontowf1o 6565

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-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

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-clab 2742  df-cleq 2755  df-clel 2839  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-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  rusgrnumwlkg  30626  f1ocnt  33488