Theorem dffv4g 5493

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 4980), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
dffv4g	⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∪ {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.

Step	Hyp	Ref	Expression
1		dffv3g 5492	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})))
2		df-iota 5160	. . 3 ⊢ (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}}
3		abid2 2291	. . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴})
4	3	eqeq1i 2178	. . . . 5 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥})
5	4	abbii 2286	. . . 4 ⊢ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
6	5	unieqi 3806	. . 3 ⊢ ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
7	2, 6	eqtri 2191	. 2 ⊢ (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
8	1, 7	eqtrdi 2219	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {cab 2156  {csn 3583  ∪ cuni 3796   “ cima 4614  ℩cio 5158  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fv 5206

This theorem is referenced by: (None)