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Mirrors > Home > MPE Home > Th. List > funfv2 | Structured version Visualization version GIF version |
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 6961 | . 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) | |
2 | funrel 6551 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
3 | relimasn 6069 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦}) |
5 | 4 | unieqd 4912 | . 2 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
6 | 1, 5 | eqtrd 2771 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 {cab 2708 {csn 4619 ∪ cuni 4898 class class class wbr 5138 “ cima 5669 Rel wrel 5671 Fun wfun 6523 ‘cfv 6529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-fv 6537 |
This theorem is referenced by: funfv2f 6963 |
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