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Theorem eufnfv 6395
 Description: A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
Hypotheses
Ref Expression
eufnfv.1 𝐴 ∈ V
eufnfv.2 𝐵 ∈ V
Assertion
Ref Expression
eufnfv ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eufnfv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eufnfv.1 . . . . 5 𝐴 ∈ V
21mptex 6390 . . . 4 (𝑥𝐴𝐵) ∈ V
3 eqeq2 2621 . . . . . 6 (𝑧 = (𝑥𝐴𝐵) → (𝑓 = 𝑧𝑓 = (𝑥𝐴𝐵)))
43bibi2d 331 . . . . 5 (𝑧 = (𝑥𝐴𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
54albidv 1836 . . . 4 (𝑧 = (𝑥𝐴𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
62, 5spcev 3273 . . 3 (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵)) → ∃𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
7 eufnfv.2 . . . . . . 7 𝐵 ∈ V
8 eqid 2610 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
97, 8fnmpti 5935 . . . . . 6 (𝑥𝐴𝐵) Fn 𝐴
10 fneq1 5893 . . . . . 6 (𝑓 = (𝑥𝐴𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
119, 10mpbiri 247 . . . . 5 (𝑓 = (𝑥𝐴𝐵) → 𝑓 Fn 𝐴)
1211pm4.71ri 663 . . . 4 (𝑓 = (𝑥𝐴𝐵) ↔ (𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)))
13 dffn5 6151 . . . . . . 7 (𝑓 Fn 𝐴𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
14 eqeq1 2614 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)) → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
1513, 14sylbi 206 . . . . . 6 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
16 fvex 6113 . . . . . . . 8 (𝑓𝑥) ∈ V
1716rgenw 2908 . . . . . . 7 𝑥𝐴 (𝑓𝑥) ∈ V
18 mpteqb 6207 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ V → ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
1917, 18ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
2015, 19syl6bb 275 . . . . 5 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2120pm5.32i 667 . . . 4 ((𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2212, 21bitr2i 264 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))
236, 22mpg 1715 . 2 𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧)
24 df-eu 2462 . 2 (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ ∃𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
2523, 24mpbir 220 1 ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  Vcvv 3173   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by: (None)
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