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| Mirrors > Home > MPE Home > Th. List > ssiun2s | Structured version Visualization version GIF version | ||
| Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.) |
| Ref | Expression |
|---|---|
| ssiun2s.1 | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
| Ref | Expression |
|---|---|
| ssiun2s | ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 2 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑥𝐷 | |
| 3 | nfiu1 5027 | . . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
| 4 | 2, 3 | nfss 3976 | . 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
| 5 | ssiun2s.1 | . . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
| 6 | 5 | sseq1d 4015 | . 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
| 7 | ssiun2 5047 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 8 | 1, 4, 6, 7 | vtoclgaf 3576 | 1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: fviunfun 7969 onfununi 8381 oaordi 8584 omordi 8604 dffi3 9471 alephordi 10114 domtriomlem 10482 pwxpndom2 10705 wunex2 10778 imasaddvallem 17574 imasvscaval 17583 iundisj2 25584 voliunlem1 25585 volsup 25591 iundisj2fi 32799 constr01 33783 bnj906 34944 bnj1137 35009 bnj1408 35050 cvmliftlem10 35299 cvmliftlem13 35301 sstotbnd2 37781 mapdrvallem3 41648 onsucunifi 43383 fvmptiunrelexplb0d 43697 fvmptiunrelexplb1d 43699 corclrcl 43720 trclrelexplem 43724 corcltrcl 43752 cotrclrcl 43755 iunincfi 45099 iundjiunlem 46474 meaiuninc3v 46499 caratheodorylem1 46541 ovnhoilem1 46616 |
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