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Theorem dffv4g 5203
 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 4722), 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 5202 . 2 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})))
2 df-iota 4895 . . 3 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3 abid2 2174 . . . . . 6 {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43eqeq1i 2063 . . . . 5 ({𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
54abbii 2169 . . . 4 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
65unieqi 3618 . . 3 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
72, 6eqtri 2076 . 2 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
81, 7syl6eq 2104 1 (𝐴𝑉 → (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   ∈ wcel 1409  {cab 2042  {csn 3403  ∪ cuni 3608   “ cima 4376  ℩cio 4893  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fv 4938 This theorem is referenced by: (None)
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