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Mirrors > Home > MPE Home > Th. List > iuneq1d | Structured version Visualization version GIF version |
Description: Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) |
Ref | Expression |
---|---|
iuneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
iuneq1d | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | iuneq1 4920 | . 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∪ ciun 4904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-v 3410 df-in 3873 df-ss 3883 df-iun 4906 |
This theorem is referenced by: iuneq12d 4932 disjxiun 5050 kmlem11 9774 prmreclem4 16472 imasval 17016 iundisj 24445 iundisj2 24446 voliunlem1 24447 iunmbl 24450 volsup 24453 uniioombllem4 24483 iuninc 30619 iundisjf 30647 iundisj2f 30648 suppovss 30737 iundisjfi 30837 iundisj2fi 30838 iundisjcnt 30839 indval2 31694 sigaclcu3 31802 fiunelros 31854 meascnbl 31899 bnj1113 32478 bnj155 32572 bnj570 32598 bnj893 32621 cvmliftlem10 32969 mrsubvrs 33197 msubvrs 33235 voliunnfl 35558 volsupnfl 35559 heiborlem3 35708 heibor 35716 iunrelexp0 40987 iunp1 42287 iundjiunlem 43672 iundjiun 43673 meaiuninclem 43693 meaiuninc 43694 carageniuncllem1 43734 carageniuncllem2 43735 carageniuncl 43736 caratheodorylem1 43739 caratheodorylem2 43740 |
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