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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 6658 | . . 3 ⊢ (𝐹‘𝑥) ∈ V | |
2 | 1 | dfiun2 4920 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
3 | fnrnfv 6700 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
4 | 3 | unieqd 4814 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
5 | 2, 4 | eqtr4id 2852 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 {cab 2776 ∃wrex 3107 ∪ cuni 4800 ∪ ciun 4881 ran crn 5520 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: funiunfv 6985 dffi3 8879 jech9.3 9227 hsmexlem5 9841 wuncval2 10158 dprdspan 19142 tgcmp 22006 txcmplem1 22246 txcmplem2 22247 xkococnlem 22264 alexsubALT 22656 bcth3 23935 ovolfioo 24071 ovolficc 24072 voliunlem2 24155 voliunlem3 24156 volsup 24160 uniiccdif 24182 uniioovol 24183 uniiccvol 24184 uniioombllem2 24187 uniioombllem4 24190 volsup2 24209 itg1climres 24318 itg2monolem1 24354 itg2gt0 24364 sigapildsys 31531 omssubadd 31668 carsgclctunlem3 31688 dftrpred2 33171 pibt2 34834 volsupnfl 35102 hbt 40074 ovolval4lem1 43288 ovolval5lem3 43293 ovnovollem1 43295 ovnovollem2 43296 |
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