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Theorem eufnfv 6969
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 6963 . . . 4 (𝑥𝐴𝐵) ∈ V
3 eqeq2 2810 . . . . . 6 (𝑧 = (𝑥𝐴𝐵) → (𝑓 = 𝑧𝑓 = (𝑥𝐴𝐵)))
43bibi2d 346 . . . . 5 (𝑧 = (𝑥𝐴𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
54albidv 1921 . . . 4 (𝑧 = (𝑥𝐴𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
62, 5spcev 3555 . . 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 2798 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
97, 8fnmpti 6463 . . . . . 6 (𝑥𝐴𝐵) Fn 𝐴
10 fneq1 6414 . . . . . 6 (𝑓 = (𝑥𝐴𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
119, 10mpbiri 261 . . . . 5 (𝑓 = (𝑥𝐴𝐵) → 𝑓 Fn 𝐴)
1211pm4.71ri 564 . . . 4 (𝑓 = (𝑥𝐴𝐵) ↔ (𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)))
13 dffn5 6699 . . . . . . 7 (𝑓 Fn 𝐴𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
14 eqeq1 2802 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)) → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
1513, 14sylbi 220 . . . . . 6 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
16 fvex 6658 . . . . . . . 8 (𝑓𝑥) ∈ V
1716rgenw 3118 . . . . . . 7 𝑥𝐴 (𝑓𝑥) ∈ V
18 mpteqb 6764 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ V → ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
1917, 18ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
2015, 19syl6bb 290 . . . . 5 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2120pm5.32i 578 . . . 4 ((𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2212, 21bitr2i 279 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))
236, 22mpg 1799 . 2 𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧)
24 eu6 2634 . 2 (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ ∃𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
2523, 24mpbir 234 1 ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2111  ∃!weu 2628  wral 3106  Vcvv 3441  cmpt 5110   Fn wfn 6319  cfv 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332
This theorem is referenced by: (None)
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