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 4898 | . . . . 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 3383 | . . 3 ⊢ (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵}) |
4 | funfn 6399 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
5 | fncnvima2 6870 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵}) | |
6 | 4, 5 | sylbi 220 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵}) |
7 | iunrab 4951 | . . . 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 6870 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) | |
11 | 4, 10 | sylbi 220 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵}) |
12 | 11 | iuneq2d 4923 | . 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 209 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 {crab 3058 ∪ ciun 4894 ◡ccnv 5539 dom cdm 5540 “ cima 5543 Fun wfun 6363 Fn wfn 6364 ‘cfv 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-fv 6377 |
This theorem is referenced by: elrspunidl 31292 |
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