Theorem maprnin 29506

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 6045	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 5892	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶))
3	2	baibr 945	. . . . . 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 669	. . . 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 7886	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶𝐵)
9	8	anbi1i 731	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶))
10		fin 6085	. . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶))
11	5, 9, 10	3bitr4ri 293	. . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶))
12	11	abbii 2739	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 4799	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 7869	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 2921	. 2 ⊢ {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2654	1 ⊢ ((𝐵 ∩ 𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ran crn 5115   Fn wfn 5883  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  fpwrelmapffs  29509