Theorem dffv4 6819

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 6041), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
dffv4	⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dffv4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffv3 6818	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴}))
2		df-iota 6437	. 2 ⊢ (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3		abid2 2868	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	eqeq1i 2736	. . . 4 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
5	4	abbii 2798	. . 3 ⊢ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
6	5	unieqi 4871	. 2 ⊢ ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
7	1, 2, 6	3eqtri 2758	1 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}

Syntax hints:   = wceq 1541   ∈ wcel 2111  {cab 2709  {csn 4576  ∪ cuni 4859   “ cima 5619  ℩cio 6435  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fv 6489

This theorem is referenced by:  bj-imafv  37284  csbfv12gALTVD  44930