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Theorem dffv4 6894
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 6096), 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 6893 . 2 (𝐹𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴}))
2 df-iota 6500 . 2 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3 abid2 2867 . . . . 5 {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43eqeq1i 2733 . . . 4 ({𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
54abbii 2798 . . 3 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
65unieqi 4920 . 2 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
71, 2, 63eqtri 2760 1 (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2705  {csn 4629   cuni 4908  cima 5681  cio 6498  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556
This theorem is referenced by:  bj-imafv  36730  csbfv12gALTVD  44338
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