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| Mirrors > Home > MPE Home > Th. List > iuneq2d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.) |
| Ref | Expression |
|---|---|
| iuneq2d.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| iuneq2d | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iuneq2d.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
| 3 | 2 | iuneq2dv 5016 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: iununi 5099 oelim2 8633 ituniiun 10462 rtrclreclem1 15096 dfrtrclrec2 15097 rtrclreclem2 15098 rtrclreclem4 15100 imasval 17556 mreacs 17701 pzriprnglem10 21501 cnextval 24069 taylfval 26400 iunpreima 32577 constrlim 33780 reprdifc 34642 msubvrs 35565 neibastop2 36362 voliunnfl 37671 sstotbnd2 37781 equivtotbnd 37785 totbndbnd 37796 heiborlem3 37820 eliunov2 43692 fvmptiunrelexplb0d 43697 fvmptiunrelexplb1d 43699 comptiunov2i 43719 trclrelexplem 43724 dftrcl3 43733 trclfvcom 43736 cnvtrclfv 43737 cotrcltrcl 43738 trclimalb2 43739 trclfvdecomr 43741 dfrtrcl3 43746 dfrtrcl4 43751 isomenndlem 46545 ovnval 46556 hoicvr 46563 hoicvrrex 46571 ovnlecvr 46573 ovncvrrp 46579 ovnsubaddlem1 46585 hoidmvlelem3 46612 hoidmvle 46615 ovnhoilem1 46616 ovnovollem1 46671 smflimlem3 46788 otiunsndisjX 47291 |
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