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Theorem funiunfvdmf 5295
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5294 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
Hypothesis
Ref Expression
funiunfvf.1 x𝐹
Assertion
Ref Expression
funiunfvdmf (𝐹 Fn A x A (𝐹x) = (𝐹A))
Distinct variable group:   x,A
Allowed substitution hint:   𝐹(x)

Proof of Theorem funiunfvdmf
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 funiunfvf.1 . . . 4 x𝐹
2 nfcv 2160 . . . 4 xz
31, 2nffv 5077 . . 3 x(𝐹z)
4 nfcv 2160 . . 3 z(𝐹x)
5 fveq2 5070 . . 3 (z = x → (𝐹z) = (𝐹x))
63, 4, 5cbviun 3646 . 2 z A (𝐹z) = x A (𝐹x)
7 funiunfvdm 5294 . 2 (𝐹 Fn A z A (𝐹z) = (𝐹A))
86, 7syl5eqr 2068 1 (𝐹 Fn A x A (𝐹x) = (𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1373  wnfc 2147   cuni 3532   ciun 3609  cima 4241   Fn wfn 4791  cfv 4796
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 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804
This theorem is referenced by: (None)
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