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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > maprnin | Structured version Visualization version GIF version |
Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.) |
Ref | Expression |
---|---|
maprnin.1 | ⊢ 𝐴 ∈ V |
maprnin.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
maprnin | ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6705 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴) | |
2 | df-f 6537 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶)) | |
3 | 2 | baibr 537 | . . . . . 6 ⊢ (𝑓 Fn 𝐴 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶)) |
4 | 1, 3 | syl 17 | . . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶)) |
5 | 4 | pm5.32i 575 | . . . 4 ⊢ ((𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶)) |
6 | maprnin.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
7 | maprnin.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
8 | 6, 7 | elmap 8850 | . . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵) |
9 | 8 | anbi1i 624 | . . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶)) |
10 | fin 6759 | . . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶)) | |
11 | 5, 9, 10 | 3bitr4ri 303 | . . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)) |
12 | 11 | abbii 2802 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)} |
13 | 6 | inex1 5311 | . . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V |
14 | 13, 7 | mapval 8817 | . 2 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} |
15 | df-rab 3433 | . 2 ⊢ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)} | |
16 | 12, 14, 15 | 3eqtr4i 2770 | 1 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 {crab 3432 Vcvv 3474 ∩ cin 3944 ⊆ wss 3945 ran crn 5671 Fn wfn 6528 ⟶wf 6529 (class class class)co 7394 ↑m cmap 8805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-map 8807 |
This theorem is referenced by: fpwrelmapffs 31894 |
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