Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 5833 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | r19.42v 3073 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥)) | |
3 | 2 | bicomi 213 | . . . . . 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 4376 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝑥) | |
6 | 5 | anbi2i 726 | . . . . . 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 6245 | . . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))) | |
9 | 8 | rexbidv 3034 | . . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ 𝑥))) |
10 | 4, 7, 9 | 3bitr4d 299 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (◡𝐹 “ 𝑥))) |
11 | elpreima 6245 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹‘𝑦) ∈ ∪ 𝐴))) | |
12 | eliun 4460 | . . . . 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 299 | . . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ (◡𝐹 “ ∪ 𝐴) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))) |
15 | 14 | eqrdv 2608 | . 2 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
16 | 1, 15 | sylbi 206 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∪ cuni 4372 ∪ ciun 4455 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: imambfm 29651 dstrvprob 29860 |
Copyright terms: Public domain | W3C validator |