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Mirrors > Home > MPE Home > Th. List > dffv4 | Structured version Visualization version GIF version |
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 6049), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
dffv4 | ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffv3 6843 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) | |
2 | df-iota 6453 | . 2 ⊢ (℩𝑦𝑦 ∈ (𝐹 “ {𝐴})) = ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} | |
3 | abid2 2876 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = (𝐹 “ {𝐴}) | |
4 | 3 | eqeq1i 2742 | . . . 4 ⊢ ({𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}) |
5 | 4 | abbii 2807 | . . 3 ⊢ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} |
6 | 5 | unieqi 4883 | . 2 ⊢ ∪ {𝑥 ∣ {𝑦 ∣ 𝑦 ∈ (𝐹 “ {𝐴})} = {𝑥}} = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} |
7 | 1, 2, 6 | 3eqtri 2769 | 1 ⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2714 {csn 4591 ∪ cuni 4870 “ cima 5641 ℩cio 6451 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-xp 5644 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fv 6509 |
This theorem is referenced by: bj-imafv 35751 csbfv12gALTVD 43255 |
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