Theorem funiunfvdmf 5346

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5345 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 2175	. . . 4 ⊢ Ⅎxz
3	1, 2	nffv 5128	. . 3 ⊢ Ⅎx(𝐹‘z)
4		nfcv 2175	. . 3 ⊢ Ⅎz(𝐹‘x)
5		fveq2 5121	. . 3 ⊢ (z = x → (𝐹‘z) = (𝐹‘x))
6	3, 4, 5	cbviun 3685	. 2 ⊢ ∪ z ∈ A (𝐹‘z) = ∪ x ∈ A (𝐹‘x)
7		funiunfvdm 5345	. 2 ⊢ (𝐹 Fn A → ∪ z ∈ A (𝐹‘z) = ∪ (𝐹 “ A))
8	6, 7	syl5eqr 2083	1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

Syntax hints:   → wi 4   = wceq 1242  Ⅎwnfc 2162  ∪ cuni 3571  ∪ ciun 3648   “ cima 4291   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)