Theorem funiunfvdmf 5716

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5715 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)

Hypothesis

Ref	Expression
funiunfvf.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
funiunfvdmf	⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem funiunfvdmf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funiunfvf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
2		nfcv 2299	. . . 4 ⊢ Ⅎ𝑥𝑧
3	1, 2	nffv 5480	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧)
4		nfcv 2299	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥)
5		fveq2 5470	. . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6	3, 4, 5	cbviun 3888	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
7		funiunfvdm 5715	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴))
8	6, 7	eqtr3id 2204	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1335  Ⅎwnfc 2286  ∪ cuni 3774  ∪ ciun 3851   “ cima 4591   Fn wfn 5167  ‘cfv 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180

This theorem is referenced by: (None)