ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eluniimadm Structured version   GIF version

Theorem eluniimadm 5317
Description: Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
eluniimadm (𝐹 Fn A → (B (𝐹A) ↔ x A B (𝐹x)))
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem eluniimadm
StepHypRef Expression
1 eliun 3624 . 2 (B x A (𝐹x) ↔ x A B (𝐹x))
2 funiunfvdm 5315 . . 3 (𝐹 Fn A x A (𝐹x) = (𝐹A))
32eleq2d 2080 . 2 (𝐹 Fn A → (B x A (𝐹x) ↔ B (𝐹A)))
41, 3syl5rbbr 184 1 (𝐹 Fn A → (B (𝐹A) ↔ x A B (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1366  wrex 2276   cuni 3543   ciun 3620  cima 4263   Fn wfn 4812  cfv 4817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-fv 4825
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator