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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unipreima | Structured version Visualization version GIF version |
Description: Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.) |
Ref | Expression |
---|---|
unipreima | ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6577 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | r19.42v 3185 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)) | |
3 | 2 | bicomi 223 | . . . . . 6 ⊢ ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥)) |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))) |
5 | eluni2 4907 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) | |
6 | 5 | anbi2i 622 | . . . . . 6 ⊢ ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)) |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥))) |
8 | elpreima 7061 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))) | |
9 | 8 | rexbidv 3173 | . . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))) |
10 | 4, 7, 9 | 3bitr4d 311 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥))) |
11 | elpreima 7061 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴))) | |
12 | eliun 4995 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥)) | |
13 | 12 | a1i 11 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥))) |
14 | 10, 11, 13 | 3bitr4d 311 | . . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))) |
15 | 14 | eqrdv 2725 | . 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
16 | 1, 15 | sylbi 216 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ∪ cuni 4903 ∪ ciun 4991 ◡ccnv 5671 dom cdm 5672 “ cima 5675 Fun wfun 6536 Fn wfn 6537 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 |
This theorem is referenced by: imambfm 33805 dstrvprob 34014 |
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