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| Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.) | 
| Ref | Expression | 
|---|---|
| funfv2 | ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funfv 6995 | . 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) | |
| 2 | funrel 6582 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
| 3 | relimasn 6102 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) | 
| 5 | 4 | unieqd 4919 | . 2 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) | 
| 6 | 1, 5 | eqtrd 2776 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 {cab 2713 {csn 4625 ∪ cuni 4906 class class class wbr 5142 “ cima 5687 Rel wrel 5689 Fun wfun 6554 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: funfv2f 6997 | 
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