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Mirrors > Home > ILE Home > Th. List > funfvdm2f | GIF version |
Description: The value of a function. Version of funfvdm2 5237 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.) |
Ref | Expression |
---|---|
funfvdm2f.1 | ⊢ Ⅎ𝑦𝐴 |
funfvdm2f.2 | ⊢ Ⅎ𝑦𝐹 |
Ref | Expression |
---|---|
funfvdm2f | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvdm2 5237 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤}) | |
2 | funfvdm2f.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | funfvdm2f.2 | . . . . 5 ⊢ Ⅎ𝑦𝐹 | |
4 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
5 | 2, 3, 4 | nfbr 3808 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤 |
6 | nfv 1421 | . . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦 | |
7 | breq2 3768 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦)) | |
8 | 5, 6, 7 | cbvab 2160 | . . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦} |
9 | 8 | unieqi 3590 | . 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦} |
10 | 1, 9 | syl6eq 2088 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {cab 2026 Ⅎwnfc 2165 ∪ cuni 3580 class class class wbr 3764 dom cdm 4345 Fun wfun 4896 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: (None) |
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