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Theorem funiunfvdmf 5435
 Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5434 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 2220 . . . 4 𝑥𝑧
31, 2nffv 5216 . . 3 𝑥(𝐹𝑧)
4 nfcv 2220 . . 3 𝑧(𝐹𝑥)
5 fveq2 5209 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
63, 4, 5cbviun 3723 . 2 𝑧𝐴 (𝐹𝑧) = 𝑥𝐴 (𝐹𝑥)
7 funiunfvdm 5434 . 2 (𝐹 Fn 𝐴 𝑧𝐴 (𝐹𝑧) = (𝐹𝐴))
86, 7syl5eqr 2128 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285  Ⅎwnfc 2207  ∪ cuni 3609  ∪ ciun 3686   “ cima 4374   Fn wfn 4927  ‘cfv 4932 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940 This theorem is referenced by: (None)
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