Theorem funiunfvf 6411

Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 6410 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	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
funiunfvf	⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem funiunfvf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funiunfvf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
2		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝑧
3	1, 2	nffv 6110	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧)
4		nfcv 2751	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥)
5		fveq2 6103	. . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
6	3, 4, 5	cbviun 4493	. 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
7		funiunfv 6410	. 2 ⊢ (Fun 𝐹 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴))
8	6, 7	syl5eqr 2658	1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1475  Ⅎwnfc 2738  ∪ cuni 4372  ∪ ciun 4455   “ cima 5041  Fun wfun 5798  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by: (None)