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Mirrors > Home > MPE Home > Th. List > funfv2f | Structured version Visualization version GIF version |
Description: The value of a function. Version of funfv2 6751 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.) |
Ref | Expression |
---|---|
funfv2f.1 | ⊢ Ⅎ𝑦𝐴 |
funfv2f.2 | ⊢ Ⅎ𝑦𝐹 |
Ref | Expression |
---|---|
funfv2f | ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfv2 6751 | . 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤}) | |
2 | funfv2f.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | funfv2f.2 | . . . . 5 ⊢ Ⅎ𝑦𝐹 | |
4 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
5 | 2, 3, 4 | nfbr 5113 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤 |
6 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦 | |
7 | breq2 5070 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦)) | |
8 | 5, 6, 7 | cbvabw 2890 | . . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦} |
9 | 8 | unieqi 4851 | . 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦} |
10 | 1, 9 | syl6eq 2872 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {cab 2799 Ⅎwnfc 2961 ∪ cuni 4838 class class class wbr 5066 Fun wfun 6349 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: (None) |
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