Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fimarab | Structured version Visualization version GIF version |
Description: Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.) |
Ref | Expression |
---|---|
fimarab | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . 2 ⊢ Ⅎ𝑦(𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) | |
2 | nfcv 2979 | . 2 ⊢ Ⅎ𝑦(𝐹 “ 𝑋) | |
3 | nfrab1 3386 | . 2 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} | |
4 | ffn 6516 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
5 | fvelimab 6739 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)) | |
6 | 5 | anbi2d 630 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))) |
7 | 4, 6 | sylan 582 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))) |
8 | fimass 6557 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) ⊆ 𝐵) |
10 | 9 | sseld 3968 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) → 𝑦 ∈ 𝐵)) |
11 | 10 | pm4.71rd 565 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)))) |
12 | rabid 3380 | . . . 4 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)) | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))) |
14 | 7, 11, 13 | 3bitr4d 313 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})) |
15 | 1, 2, 3, 14 | eqrd 3988 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 {crab 3144 ⊆ wss 3938 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 |
This theorem is referenced by: locfinreflem 31106 |
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