Theorem f1ofveu 5830

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 5445	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1of 5432	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
3	1, 2	syl 14	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵⟶𝐴)
4		feu 5370	. . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
5	3, 4	sylan 281	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
6		f1ocnvfvb 5748	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
7	6	3com23 1199	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
8		dff1o4 5440	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	8	simprbi 273	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹 Fn 𝐵)
10		fnopfvb 5528	. . . . . . 7 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
11	10	3adant3 1007	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
12	9, 11	syl3an1 1261	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
13	7, 12	bitrd 187	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
14	13	3expa 1193	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
15	14	reubidva 2648	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
16	5, 15	mpbird 166	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∃!wreu 2446  ⟨cop 3579  ^◡ccnv 4603   Fn wfn 5183  ⟶wf 5184  –1-1-onto→wf1o 5187  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  1arith2  12298