Theorem f1ofveu 7442

Description: There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)

Assertion

Ref	Expression
f1ofveu	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem f1ofveu

Step	Hyp	Ref	Expression
1		f1ocnv 6874	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1of 6862	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵⟶𝐴)
4		feu 6797	. . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
5	3, 4	sylan 579	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
6		f1ocnvfvb 7315	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
7	6	3com23 1126	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
8		dff1o4 6870	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	8	simprbi 496	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹 Fn 𝐵)
10		fnopfvb 6974	. . . . . . 7 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
11	10	3adant3 1132	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
12	9, 11	syl3an1 1163	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
13	7, 12	bitrd 279	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
14	13	3expa 1118	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
15	14	reubidva 3404	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
16	5, 15	mpbird 257	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃!wreu 3386  ⟨cop 4654  ^◡ccnv 5699   Fn wfn 6568  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  1arith2  16975  uspgredgiedg  29210  disjrdx  32613  ply1divalg3  35610  reuf1odnf  47022  reuf1od  47023