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Mirrors > Home > NFE Home > Th. List > dfiun2 | GIF version |
Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
dfiun2.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
dfiun2 | ⊢ ∪x ∈ A B = ∪{y ∣ ∃x ∈ A y = B} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun2g 3999 | . 2 ⊢ (∀x ∈ A B ∈ V → ∪x ∈ A B = ∪{y ∣ ∃x ∈ A y = B}) | |
2 | dfiun2.1 | . . 3 ⊢ B ∈ V | |
3 | 2 | a1i 10 | . 2 ⊢ (x ∈ A → B ∈ V) |
4 | 1, 3 | mprg 2683 | 1 ⊢ ∪x ∈ A B = ∪{y ∣ ∃x ∈ A y = B} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2615 Vcvv 2859 ∪cuni 3891 ∪ciun 3969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-uni 3892 df-iun 3971 |
This theorem is referenced by: funcnvuni 5161 fun11iun 5305 fniunfv 5466 |
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