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Mirrors > Home > ILE Home > Th. List > funfvdm2f | GIF version |
Description: The value of a function. Version of funfvdm2 5534 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 5534 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤}) | |
2 | funfvdm2f.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | funfvdm2f.2 | . . . . 5 ⊢ Ⅎ𝑦𝐹 | |
4 | nfcv 2299 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
5 | 2, 3, 4 | nfbr 4012 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤 |
6 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦 | |
7 | breq2 3971 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦)) | |
8 | 5, 6, 7 | cbvab 2281 | . . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦} |
9 | 8 | unieqi 3784 | . 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦} |
10 | 1, 9 | eqtrdi 2206 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 {cab 2143 Ⅎwnfc 2286 ∪ cuni 3774 class class class wbr 3967 dom cdm 4588 Fun wfun 5166 ‘cfv 5172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-fv 5180 |
This theorem is referenced by: (None) |
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