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Theorem funiunfvdmf 5781
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5780 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 𝑥𝐴 (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 2332 . . . 4 𝑥𝑧
31, 2nffv 5540 . . 3 𝑥(𝐹𝑧)
4 nfcv 2332 . . 3 𝑧(𝐹𝑥)
5 fveq2 5530 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
63, 4, 5cbviun 3938 . 2 𝑧𝐴 (𝐹𝑧) = 𝑥𝐴 (𝐹𝑥)
7 funiunfvdm 5780 . 2 (𝐹 Fn 𝐴 𝑧𝐴 (𝐹𝑧) = (𝐹𝐴))
86, 7eqtr3id 2236 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wnfc 2319   cuni 3824   ciun 3901  cima 4644   Fn wfn 5226  cfv 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239
This theorem is referenced by: (None)
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