Theorem maprnin 33205

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 6703	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
2		df-f 6537	. . . . . . 7 ⊢ (𝑓:𝐴⟶𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓 ⊆ 𝐶))
3	2	baibr 546	. . . . . 6 ⊢ (𝑓 Fn 𝐴 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶))
4	1, 3	syl 18	. . . . 5 ⊢ (𝑓:𝐴⟶𝐵 → (ran 𝑓 ⊆ 𝐶 ↔ 𝑓:𝐴⟶𝐶))
5	4	pm5.32i 585	. . . 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 8881	. . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐵)
9	8	anbi1i 636	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ ran 𝑓 ⊆ 𝐶))
10		fin 6756	. . . 4 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓:𝐴⟶𝐵 ∧ 𝑓:𝐴⟶𝐶))
11	5, 9, 10	3bitr4ri 307	. . 3 ⊢ (𝑓:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶))
12	11	abbii 2827	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
13	6	inex1 5280	. . 3 ⊢ (𝐵 ∩ 𝐶) ∈ V
14	13, 7	mapval 8840	. 2 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶(𝐵 ∩ 𝐶)}
15		df-rab 3413	. 2 ⊢ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵 ↑m 𝐴) ∧ ran 𝑓 ⊆ 𝐶)}
16	12, 14, 15	3eqtr4i 2793	1 ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ran crn 5656   Fn wfn 6528  ⟶wf 6529  (class class class)co 7414   ↑_m cmap 8829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8831

This theorem is used by:  fpwrelmapffs  33208