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Theorem dffv4g 5175
 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 4694), 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 5174 . 2 (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})))
2 df-iota 4867 . . 3 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3 abid2 2158 . . . . . 6 {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
43eqeq1i 2047 . . . . 5 ({𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
54abbii 2153 . . . 4 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
65unieqi 3590 . . 3 {𝑥 ∣ {𝑦𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
72, 6eqtri 2060 . 2 (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
81, 7syl6eq 2088 1 (𝐴𝑉 → (𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  {cab 2026  {csn 3375  ∪ cuni 3580   “ cima 4348  ℩cio 4865  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fv 4910 This theorem is referenced by: (None)
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