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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 5037 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
3 | fnrnfv 6967 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
4 | 3 | unieqd 4924 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
5 | 2, 4 | eqtr4id 2793 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {cab 2711 ∃wrex 3067 ∪ cuni 4911 ∪ ciun 4995 ran crn 5689 Fn wfn 6557 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: funiunfv 7267 dffi3 9468 jech9.3 9851 hsmexlem5 10467 wuncval2 10784 dprdspan 20061 tgcmp 23424 txcmplem1 23664 txcmplem2 23665 xkococnlem 23682 alexsubALT 24074 bcth3 25378 ovolfioo 25515 ovolficc 25516 voliunlem2 25599 voliunlem3 25600 volsup 25604 uniiccdif 25626 uniioovol 25627 uniiccvol 25628 uniioombllem2 25631 uniioombllem4 25634 volsup2 25653 itg1climres 25763 itg2monolem1 25799 itg2gt0 25809 sigapildsys 34142 omssubadd 34281 carsgclctunlem3 34301 pibt2 37399 volsupnfl 37651 hbt 43118 ovolval4lem1 46604 ovolval5lem3 46609 ovnovollem1 46611 ovnovollem2 46612 |
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