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Theorem dffv4 6903
Description: The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 6110), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
dffv4 (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dffv4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dffv3 6902 . 2 (𝐹𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴}))
2 df-iota 6514 . 2 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3 abid2 2879 . . . . 5 {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43eqeq1i 2742 . . . 4 ({𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
54abbii 2809 . . 3 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
65unieqi 4919 . 2 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
71, 2, 63eqtri 2769 1 (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  {csn 4626   cuni 4907  cima 5688  cio 6512  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569
This theorem is referenced by:  bj-imafv  37252  csbfv12gALTVD  44919
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