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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 6305 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 6305 | . 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤}) | |
2 | funfv2f.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | funfv2f.2 | . . . . 5 ⊢ Ⅎ𝑦𝐹 | |
4 | nfcv 2793 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
5 | 2, 3, 4 | nfbr 4732 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤 |
6 | nfv 1883 | . . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦 | |
7 | breq2 4689 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦)) | |
8 | 5, 6, 7 | cbvab 2775 | . . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦} |
9 | 8 | unieqi 4477 | . 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦} |
10 | 1, 9 | syl6eq 2701 | 1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 {cab 2637 Ⅎwnfc 2780 ∪ cuni 4468 class class class wbr 4685 Fun wfun 5920 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 |
This theorem is referenced by: (None) |
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