Theorem f1ofveu 5520

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 5159	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1of 5146	. . . 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 5092	. . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
5	3, 4	sylan 277	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
6		f1ocnvfvb 5440	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
7	6	3com23 1144	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
8		dff1o4 5154	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	8	simprbi 269	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹 Fn 𝐵)
10		fnopfvb 5236	. . . . . . 7 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
11	10	3adant3 958	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
12	9, 11	syl3an1 1202	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
13	7, 12	bitrd 186	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
14	13	3expa 1138	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
15	14	reubidva 2536	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
16	5, 15	mpbird 165	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃!wreu 2350  ⟨cop 3401  ^◡ccnv 4362   Fn wfn 4917  ⟶wf 4918  –1-1-onto→wf1o 4921  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by: (None)