Theorem maprnin 32745

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 6747	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 6577	. . . . . . 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 8929	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵)
9	8	anbi1i 623	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶))
10		fin 6801	. . . 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 2812	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 5335	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 8896	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 3444	. 2 ⊢ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2778	1 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ran crn 5701   Fn wfn 6568  ⟶wf 6569  (class class class)co 7448   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886

This theorem is referenced by:  fpwrelmapffs  32748