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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunpreima | Structured version Visualization version GIF version |
Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
Ref | Expression |
---|---|
iunpreima | ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 4999 | . . . . 5 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (Fun 𝐹 → ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵)) |
3 | 2 | rabbidv 3440 | . . 3 ⊢ (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵}) |
4 | funfn 6597 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
5 | fncnvima2 7080 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵}) | |
6 | 4, 5 | sylbi 217 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵}) |
7 | iunrab 5056 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵} | |
8 | 7 | a1i 11 | . . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵}) |
9 | 3, 6, 8 | 3eqtr4d 2784 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) |
10 | fncnvima2 7080 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) | |
11 | 4, 10 | sylbi 217 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) |
12 | 11 | iuneq2d 5026 | . 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) |
13 | 9, 12 | eqtr4d 2777 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {crab 3432 ∪ ciun 4995 ◡ccnv 5687 dom cdm 5688 “ cima 5691 Fun wfun 6556 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-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-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: elrspunidl 33435 |
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