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Theorem funiunfvf 7197
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 7196 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)
Hypothesis
Ref Expression
funiunfvf.1 𝑥𝐹
Assertion
Ref Expression
funiunfvf (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem funiunfvf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 funiunfvf.1 . . . 4 𝑥𝐹
2 nfcv 2904 . . . 4 𝑥𝑧
31, 2nffv 6853 . . 3 𝑥(𝐹𝑧)
4 nfcv 2904 . . 3 𝑧(𝐹𝑥)
5 fveq2 6843 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
63, 4, 5cbviun 4997 . 2 𝑧𝐴 (𝐹𝑧) = 𝑥𝐴 (𝐹𝑥)
7 funiunfv 7196 . 2 (Fun 𝐹 𝑧𝐴 (𝐹𝑧) = (𝐹𝐴))
86, 7eqtr3id 2787 1 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wnfc 2884   cuni 4866   ciun 4955  cima 5637  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by: (None)
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