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Mirrors > Home > MPE Home > Th. List > iunrab | Structured version Visualization version GIF version |
Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
iunrab | ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunab 4966 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} | |
2 | df-rab 3144 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}) |
4 | 3 | iuneq2i 4931 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} |
5 | df-rab 3144 | . . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)} | |
6 | r19.42v 3347 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) | |
7 | 6 | abbii 2883 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)} |
8 | 5, 7 | eqtr4i 2844 | . 2 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} |
9 | 1, 4, 8 | 3eqtr4i 2851 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 ∃wrex 3136 {crab 3139 ∪ ciun 4910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-iun 4912 |
This theorem is referenced by: hashrabrex 15168 incexc2 15181 phisum 16115 itg2monolem1 24278 aannenlem1 24844 musum 25695 lgsquadlem1 25883 lgsquadlem2 25884 edglnl 26855 iunpreima 30244 poimirlem27 34800 cnambfre 34821 mapdval3N 38647 mapdval5N 38649 fiphp3d 39294 |
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