Theorem f1ofveu 6005

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 5596	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		f1of 5583	. . . 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 5519	. . 3 ⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
5	3, 4	sylan 283	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹)
6		f1ocnvfvb 5920	. . . . . 6 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
7	6	3com23 1235	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ (◡𝐹‘𝐶) = 𝑥))
8		dff1o4 5591	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	8	simprbi 275	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹 Fn 𝐵)
10		fnopfvb 5685	. . . . . . 7 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
11	10	3adant3 1043	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
12	9, 11	syl3an1 1306	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((◡𝐹‘𝐶) = 𝑥 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
13	7, 12	bitrd 188	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
14	13	3expa 1229	. . 3 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝐶 ↔ ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
15	14	reubidva 2717	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶 ↔ ∃!𝑥 ∈ 𝐴 ⟨𝐶, 𝑥⟩ ∈ ◡𝐹))
16	5, 15	mpbird 167	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∃!wreu 2512  ⟨cop 3672  ^◡ccnv 4724   Fn wfn 5321  ⟶wf 5322  –1-1-onto→wf1o 5325  ‘cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  1arith2  12940  uspgredgiedg  16028