Theorem dffv4 6662

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 5952), 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 6661	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴}))
2		df-iota 6309	. 2 ⊢ (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3		abid2 2957	. . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	eqeq1i 2826	. . . 4 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
5	4	abbii 2886	. . 3 ⊢ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
6	5	unieqi 4841	. 2 ⊢ ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
7	1, 2, 6	3eqtri 2848	1 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}

Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  {csn 4561  ∪ cuni 4832   “ cima 5553  ℩cio 6307  ‘cfv 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fv 6358

This theorem is referenced by:  bj-imafv  34527  csbfv12gALTVD  41226