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Mirrors > Home > ILE Home > Th. List > funfvdm2f | GIF version |
Description: The value of a function. Version of funfvdm2 5564 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 5564 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤}) | |
2 | funfvdm2f.1 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | funfvdm2f.2 | . . . . 5 ⊢ Ⅎ𝑦𝐹 | |
4 | nfcv 2313 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
5 | 2, 3, 4 | nfbr 4036 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤 |
6 | nfv 1522 | . . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦 | |
7 | breq2 3994 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦)) | |
8 | 5, 6, 7 | cbvab 2295 | . . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦} |
9 | 8 | unieqi 3807 | . 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦} |
10 | 1, 9 | eqtrdi 2220 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1349 ∈ wcel 2142 {cab 2157 Ⅎwnfc 2300 ∪ cuni 3797 class class class wbr 3990 dom cdm 4612 Fun wfun 5194 ‘cfv 5200 |
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 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-14 2145 ax-ext 2153 ax-sep 4108 ax-pow 4161 ax-pr 4195 |
This theorem depends on definitions: df-bi 116 df-3an 976 df-tru 1352 df-nf 1455 df-sb 1757 df-eu 2023 df-mo 2024 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-ral 2454 df-rex 2455 df-v 2733 df-sbc 2957 df-un 3126 df-in 3128 df-ss 3135 df-pw 3569 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-br 3991 df-opab 4052 df-id 4279 df-xp 4618 df-rel 4619 df-cnv 4620 df-co 4621 df-dm 4622 df-rn 4623 df-res 4624 df-ima 4625 df-iota 5162 df-fun 5202 df-fn 5203 df-fv 5208 |
This theorem is referenced by: (None) |
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