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 5958 | . . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴) | |
2 | df-f 5808 | . . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶)) | |
3 | 2 | baibr 943 | . . . . . 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 667 | . . . 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 7772 | . . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝐵) |
9 | 8 | anbi1i 727 | . . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶)) |
10 | fin 5998 | . . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶)) | |
11 | 5, 9, 10 | 3bitr4ri 292 | . . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)) |
12 | 11 | abbii 2726 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)} |
13 | 6 | inex1 4727 | . . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V |
14 | 13, 7 | mapval 7756 | . 2 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} |
15 | df-rab 2905 | . 2 ⊢ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)} | |
16 | 12, 14, 15 | 3eqtr4i 2642 | 1 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ran crn 5039 Fn wfn 5799 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: fpwrelmapffs 28897 |
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