Theorem funiunfvdmf 5888

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5887 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 2372	. . . 4 ⊢ Ⅎ𝑥𝑧
3	1, 2	nffv 5637	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧)
4		nfcv 2372	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥)
5		fveq2 5627	. . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6	3, 4, 5	cbviun 4002	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
7		funiunfvdm 5887	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴))
8	6, 7	eqtr3id 2276	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1395  Ⅎwnfc 2359  ∪ cuni 3888  ∪ ciun 3965   “ cima 4722   Fn wfn 5313  ‘cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)