Theorem maprnin 32654

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 6688	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 6515	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶))
3	2	baibr 536	. . . . . 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 574	. . . 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 8844	. . . . 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 6740	. . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶))
11	5, 9, 10	3bitr4ri 304	. . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶))
12	11	abbii 2796	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 5272	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 8811	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 3406	. 2 ⊢ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2762	1 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3405  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ran crn 5639   Fn wfn 6506  ⟶wf 6507  (class class class)co 7387   ↑_m cmap 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801

This theorem is referenced by:  fpwrelmapffs  32657