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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > intimafv | Structured version Visualization version GIF version |
Description: The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.) |
Ref | Expression |
---|---|
intimafv | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfimafn 6709 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}) | |
2 | 1 | inteqd 4836 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}) |
3 | fvex 6664 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
4 | 3 | rgenw 3080 | . . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V |
5 | iinabrex 30415 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
7 | eqcom 2766 | . . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥)) | |
8 | 7 | rexbii 3173 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
9 | 8 | abbii 2824 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
10 | 9 | inteqi 4835 | . . 3 ⊢ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} |
11 | 6, 10 | eqtr4i 2785 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} |
12 | 2, 11 | eqtr4di 2812 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 {cab 2736 ∀wral 3068 ∃wrex 3069 Vcvv 3407 ⊆ wss 3854 ∩ cint 4831 ∩ ciin 4877 dom cdm 5517 “ cima 5520 Fun wfun 6322 ‘cfv 6328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-int 4832 df-iin 4879 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-fv 6336 |
This theorem is referenced by: zarclsint 31328 zarcmplem 31337 |
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