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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 2955 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑥𝐷 | |
3 | nfiu1 4915 | . . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
4 | 2, 3 | nfss 3907 | . 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
5 | ssiun2s.1 | . . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
6 | 5 | sseq1d 3946 | . 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
7 | ssiun2 4934 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
8 | 1, 4, 6, 7 | vtoclgaf 3521 | 1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ ciun 4881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-in 3888 df-ss 3898 df-iun 4883 |
This theorem is referenced by: fviunfun 7628 onfununi 7961 oaordi 8155 omordi 8175 dffi3 8879 alephordi 9485 domtriomlem 9853 pwxpndom2 10076 wunex2 10149 imasaddvallem 16794 imasvscaval 16803 iundisj2 24153 voliunlem1 24154 volsup 24160 iundisj2fi 30546 bnj906 32312 bnj1137 32377 bnj1408 32418 cvmliftlem10 32654 cvmliftlem13 32656 sstotbnd2 35212 mapdrvallem3 38942 fvmptiunrelexplb0d 40385 fvmptiunrelexplb1d 40387 corclrcl 40408 trclrelexplem 40412 corcltrcl 40440 cotrclrcl 40443 iunincfi 41730 iundjiunlem 43098 meaiuninc3v 43123 caratheodorylem1 43165 ovnhoilem1 43240 |
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