Theorem fniunfv 7240

Description: The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Assertion

Ref	Expression
fniunfv	⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniunfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6887	. . 3 ⊢ (𝐹‘𝑥) ∈ V
2	1	dfiun2 4990	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
3		fnrnfv 6933	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
4	3	unieqd 4880	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
5	2, 4	eqtr4id 2814	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2738  ∃wrex 3086  ∪ cuni 4867  ∪ ciun 4951  ran crn 5649   Fn wfn 6523  ‘cfv 6528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536

This theorem is used by:  funiunfv  7241  dffi3  9401  jech9.3  9796  hsmexlem5  10465  wuncval2  10789  dprdspan  20190  tgcmp  23666  txcmplem1  23907  txcmplem2  23908  xkococnlem  23925  alexsubALT  24317  bcth3  25599  ovolfioo  25735  ovolficc  25736  voliunlem2  25819  voliunlem3  25820  volsup  25824  uniiccdif  25846  uniioovol  25847  uniiccvol  25848  uniioombllem2  25851  uniioombllem4  25854  volsup2  25873  itg1climres  25982  itg2monolem1  26018  itg2gt0  26028  sigapildsys  34714  omssubadd  34852  carsgclctunlem3  34872  pibt2  38254  volsupnfl  38497  hbt  44069  ovolval4lem1  47575  ovolval5lem3  47580  ovnovollem1  47582  ovnovollem2  47583