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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 6717 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴) | |
2 | df-f 6547 | . . . . . . 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 8864 | . . . . 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 6771 | . . . 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 5317 | . . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V |
14 | 13, 7 | mapval 8831 | . 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 3947 ⊆ wss 3948 ran crn 5677 Fn wfn 6538 ⟶wf 6539 (class class class)co 7408 ↑m cmap 8819 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
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 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 |
This theorem is referenced by: fpwrelmapffs 31954 |
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