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Mirrors > Home > MPE Home > Th. List > dfiun2 | Structured version Visualization version 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 5034 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
2 | dfiun2.1 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V) |
4 | 1, 3 | mprg 3068 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 Vcvv 3475 ∪ cuni 4909 ∪ ciun 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-uni 4910 df-iun 5000 |
This theorem is referenced by: fniunfv 7246 funcnvuni 7922 fiun 7929 f1iun 7930 tfrlem8 8384 rdglim2a 8433 rankuni 9858 cardiun 9977 kmlem11 10155 cfslb2n 10263 enfin2i 10316 pwcfsdom 10578 rankcf 10772 tskuni 10778 discmp 22902 cmpsublem 22903 cmpsub 22904 nnoeomeqom 42062 |
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