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Theorem dffv4g 5531
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 5015), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
dffv4g (𝐴𝑉 → (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑉

Proof of Theorem dffv4g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dffv3g 5530 . 2 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})))
2 df-iota 5196 . . 3 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3 abid2 2310 . . . . . 6 {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43eqeq1i 2197 . . . . 5 ({𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
54abbii 2305 . . . 4 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
65unieqi 3834 . . 3 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
72, 6eqtri 2210 . 2 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
81, 7eqtrdi 2238 1 (𝐴𝑉 → (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  {cab 2175  {csn 3607   cuni 3824  cima 4647  cio 5194  cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fv 5243
This theorem is referenced by: (None)
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