Theorem toponss 21251

Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
toponss	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Proof of Theorem toponss

Step	Hyp	Ref	Expression
1		elssuni 4737	. . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2	1	adantl 474	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ∪ 𝐽)
3		toponuni 21238	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
4	3	adantr 473	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝑋 = ∪ 𝐽)
5	2, 4	sseqtr4d 3892	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ⊆ wss 3823  ∪ cuni 4708  ‘cfv 6185  TopOnctopon 21234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-topon 21235

This theorem is referenced by:  en2top  21309  neiptopreu  21457  iscnp3  21568  cnntr  21599  cncnp  21604  isreg2  21701  connsub  21745  iunconnlem  21751  conncompclo  21759  1stccnp  21786  kgenidm  21871  tx1cn  21933  tx2cn  21934  xkoccn  21943  txcnp  21944  ptcnplem  21945  xkoinjcn  22011  idqtop  22030  qtopss  22039  kqfvima  22054  kqsat  22055  kqreglem1  22065  kqreglem2  22066  qtopf1  22140  fbflim  22300  flimcf  22306  flimrest  22307  isflf  22317  fclscf  22349  subgntr  22430  ghmcnp  22438  qustgpopn  22443  qustgplem  22444  tsmsxplem1  22476  tsmsxp  22478  ressusp  22589  mopnss  22771  xrtgioo  23129  lebnumlem2  23281  cfilfcls  23592  iscmet3lem2  23610  dvres3a  24227  dvmptfsum  24287  dvcnvlem  24288  dvcnv  24289  efopn  24954  txomap  30771  cnllysconn  32106  cvmlift2lem9a  32164  icccncfext  41625  dvmptconst  41654  dvmptidg  41656  qndenserrnopnlem  42038  opnvonmbllem2  42371