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Theorem eufnfv 7179
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 7173 . . . 4 (𝑥𝐴𝐵) ∈ V
3 eqeq2 2748 . . . . . 6 (𝑧 = (𝑥𝐴𝐵) → (𝑓 = 𝑧𝑓 = (𝑥𝐴𝐵)))
43bibi2d 342 . . . . 5 (𝑧 = (𝑥𝐴𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
54albidv 1923 . . . 4 (𝑧 = (𝑥𝐴𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))))
62, 5spcev 3565 . . 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 2736 . . . . . . 7 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
97, 8fnmpti 6644 . . . . . 6 (𝑥𝐴𝐵) Fn 𝐴
10 fneq1 6593 . . . . . 6 (𝑓 = (𝑥𝐴𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
119, 10mpbiri 257 . . . . 5 (𝑓 = (𝑥𝐴𝐵) → 𝑓 Fn 𝐴)
1211pm4.71ri 561 . . . 4 (𝑓 = (𝑥𝐴𝐵) ↔ (𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)))
13 dffn5 6901 . . . . . . 7 (𝑓 Fn 𝐴𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
14 eqeq1 2740 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)) → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
1513, 14sylbi 216 . . . . . 6 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵)))
16 fvex 6855 . . . . . . . 8 (𝑓𝑥) ∈ V
1716rgenw 3068 . . . . . . 7 𝑥𝐴 (𝑓𝑥) ∈ V
18 mpteqb 6967 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ V → ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
1917, 18ax-mp 5 . . . . . 6 ((𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
2015, 19bitrdi 286 . . . . 5 (𝑓 Fn 𝐴 → (𝑓 = (𝑥𝐴𝐵) ↔ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2120pm5.32i 575 . . . 4 ((𝑓 Fn 𝐴𝑓 = (𝑥𝐴𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵))
2212, 21bitr2i 275 . . 3 ((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = (𝑥𝐴𝐵))
236, 22mpg 1799 . 2 𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧)
24 eu6 2572 . 2 (∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ ∃𝑧𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵) ↔ 𝑓 = 𝑧))
2523, 24mpbir 230 1 ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  wral 3064  Vcvv 3445  cmpt 5188   Fn wfn 6491  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by: (None)
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