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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 4991 | . 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 3067 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3444 ∪ cuni 4866 ∪ ciun 4955 |
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 3062 df-rex 3071 df-v 3446 df-uni 4867 df-iun 4957 |
This theorem is referenced by: fniunfv 7195 funcnvuni 7869 fiun 7876 f1iun 7877 tfrlem8 8331 rdglim2a 8380 rankuni 9804 cardiun 9923 kmlem11 10101 cfslb2n 10209 enfin2i 10262 pwcfsdom 10524 rankcf 10718 tskuni 10724 discmp 22765 cmpsublem 22766 cmpsub 22767 nnoeomeqom 41690 |
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