Theorem mapsspm 8156

Description: Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)

Assertion

Ref	Expression
mapsspm	⊢ (𝐴 ↑𝑚 𝐵) ⊆ (𝐴 ↑pm 𝐵)

Proof of Theorem mapsspm

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapex 8143	. . . 4 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simprd 491	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝐵 ∈ V)
3	1	simpld 490	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝐴 ∈ V)
4		elmapi 8144	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝑓:𝐵⟶𝐴)
5		fpmg 8148	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝑓 ∈ (𝐴 ↑pm 𝐵))
6	2, 3, 4, 5	syl3anc 1494	. 2 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝑓 ∈ (𝐴 ↑pm 𝐵))
7	6	ssriv 3831	1 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ (𝐴 ↑pm 𝐵)

Syntax hints:   ∈ wcel 2164  Vcvv 3414   ⊆ wss 3798  ⟶wf 6119  (class class class)co 6905   ↑_𝑚 cmap 8122   ↑_pm cpm 8123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-pm 8125

This theorem is referenced by:  mapsspw  8158  wunmap  9863  dvntaylp  24524  taylthlem1  24526  taylthlem2  24527  mrsubrn  31945  mrsubff1  31946  msubrn  31961  msubff1  31988