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Mirrors > Home > MPE Home > Th. List > eluniima | Structured version Visualization version GIF version |
Description: Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
eluniima | ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funiunfv 7267 | . . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) | |
2 | 1 | eleq2d 2824 | . 2 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ 𝐵 ∈ ∪ (𝐹 “ 𝐴))) |
3 | eliun 4999 | . 2 ⊢ (𝐵 ∈ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥)) | |
4 | 2, 3 | bitr3di 286 | 1 ⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2105 ∃wrex 3067 ∪ cuni 4911 ∪ ciun 4995 “ cima 5691 Fun wfun 6556 ‘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-in 3969 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-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 |
This theorem is referenced by: elunirnALT 7271 ttrclse 9764 alephfp 10145 acsficl2d 18609 elhf 36155 |
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