Theorem ssiun2 5051

Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun2	⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem ssiun2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rspe 3247	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2	1	ex 414	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
3		eliun 5002	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	2, 3	imbitrrdi 251	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
5	4	ssrdv 3989	1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2107  ∃wrex 3071   ⊆ wss 3949  ∪ ciun 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-iun 5000

This theorem is referenced by:  ssiun2s  5052  disjxiun  5146  triun  5281  iunopeqop  5522  ixpf  8914  ixpiunwdom  9585  r1sdom  9769  r1val1  9781  rankuni2b  9848  rankval4  9862  cplem1  9884  domtriomlem  10437  ac6num  10474  iunfo  10534  iundom2g  10535  pwfseqlem3  10655  inar1  10770  tskuni  10778  iunconnlem  22931  ptclsg  23119  ovoliunlem1  25019  limciun  25411  ssiun2sf  31791  djussxp2  31873  suppovss  31906  bnj906  33941  bnj999  33969  bnj1014  33972  bnj1408  34047  rdgssun  36259  cpcolld  43017  iunmapss  43914  ssmapsn  43915  sge0iunmpt  45134  sge0iun  45135  voliunsge0lem  45188  omeiunltfirp  45235