maprnin - Mathbox for Thierry Arnoux

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Theorem maprnin 32534

Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)

**Hypotheses**
Ref	Expression
maprnin.1	⊢ 𝐴 ∈ V
maprnin.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
maprnin	⊢ ((𝐵 ∩ 𝐶) ↑_m 𝐴) = {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓 𝐶,𝑓

**Proof of Theorem maprnin**
Step	Hyp	Ref	Expression
1		ffn 6727	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 6557	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶))
3	2	baibr 535	. . . . . 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 573	. . . 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 8896	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐴) ↔ 𝑓:𝐴⟶𝐵)
9	8	anbi1i 622	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶))
10		fin 6782	. . . 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 2798	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 5321	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 8863	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑_m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 3431	. 2 ⊢ {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2766	1 ⊢ ((𝐵 ∩ 𝐶) ↑_m 𝐴) = {𝑓 ∈ (𝐵 ↑_m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2705 {crab 3430 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 ran crn 5683 Fn wfn 6548 ⟶wf 6549 (class class class)co 7426 ↑_m cmap 8851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853

This theorem is referenced by: fpwrelmapffs 32537

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