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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 2979 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | nfcv 2979 | . . 3 ⊢ Ⅎ𝑥𝐷 | |
3 | nfiu1 4955 | . . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
4 | 2, 3 | nfss 3962 | . 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
5 | ssiun2s.1 | . . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
6 | 5 | sseq1d 4000 | . 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
7 | ssiun2 4973 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
8 | 1, 4, 6, 7 | vtoclgaf 3575 | 1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ ciun 4921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-in 3945 df-ss 3954 df-iun 4923 |
This theorem is referenced by: fviunfun 7648 onfununi 7980 oaordi 8174 omordi 8194 dffi3 8897 alephordi 9502 domtriomlem 9866 pwxpndom2 10089 wunex2 10162 imasaddvallem 16804 imasvscaval 16813 iundisj2 24152 voliunlem1 24153 volsup 24159 iundisj2fi 30522 bnj906 32204 bnj1137 32269 bnj1408 32310 cvmliftlem10 32543 cvmliftlem13 32545 sstotbnd2 35054 mapdrvallem3 38784 fvmptiunrelexplb0d 40036 fvmptiunrelexplb1d 40038 corclrcl 40059 trclrelexplem 40063 corcltrcl 40091 cotrclrcl 40094 iunincfi 41367 iundjiunlem 42748 meaiuninc3v 42773 caratheodorylem1 42815 ovnhoilem1 42890 |
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