Theorem dffv3g 5671

Description: A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dffv3g	⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝑉

Proof of Theorem dffv3g

Step	Hyp	Ref	Expression
1		df-fv 5365	. 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		vex 2818	. . . 4 ⊢ 𝑥 ∈ V
3		elimasng 5135	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
4		df-br 4115	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
5	3, 4	bitr4di 198	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
6	2, 5	mpan2 425	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
7	6	iotabidv 5340	. 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
8	1, 7	eqtr4id 2286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815  {csn 3694  ⟨cop 3697   class class class wbr 4114   “ cima 4757  ℩cio 5315  ‘cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fv 5365

This theorem is referenced by:  dffv4g  5672  fvco2  5751  shftval  11535