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| Mirrors > Home > MPE Home > Th. List > fniunfv | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| fniunfv | ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvex 6919 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
| 2 | 1 | dfiun2 5033 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
| 3 | fnrnfv 6968 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
| 4 | 3 | unieqd 4920 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
| 5 | 2, 4 | eqtr4id 2796 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {cab 2714 ∃wrex 3070 ∪ cuni 4907 ∪ ciun 4991 ran crn 5686 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: funiunfv 7268 dffi3 9471 jech9.3 9854 hsmexlem5 10470 wuncval2 10787 dprdspan 20047 tgcmp 23409 txcmplem1 23649 txcmplem2 23650 xkococnlem 23667 alexsubALT 24059 bcth3 25365 ovolfioo 25502 ovolficc 25503 voliunlem2 25586 voliunlem3 25587 volsup 25591 uniiccdif 25613 uniioovol 25614 uniiccvol 25615 uniioombllem2 25618 uniioombllem4 25621 volsup2 25640 itg1climres 25749 itg2monolem1 25785 itg2gt0 25795 sigapildsys 34163 omssubadd 34302 carsgclctunlem3 34322 pibt2 37418 volsupnfl 37672 hbt 43142 ovolval4lem1 46664 ovolval5lem3 46669 ovnovollem1 46671 ovnovollem2 46672 |
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