Theorem mptpreima 4958

Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypothesis

Ref	Expression
dmmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
mptpreima	⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mptpreima

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmmpt2.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		df-mpt 3923	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	1, 2	eqtri 2115	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 4642	. . . 4 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 4866	. . . 4 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2115	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	imaeq1i 4804	. 2 ⊢ (◡𝐹 “ 𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶)
8		df-ima 4480	. . 3 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶)
9		resopab 4789	. . . . 5 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
10	9	rneqi 4695	. . . 4 ⊢ ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
11		ancom 263	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶))
12		anass 394	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
13	11, 12	bitri 183	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
14	13	exbii 1548	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
15		19.42v 1841	. . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
16		df-clel 2091	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))
17	16	bicomi 131	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶)
18	17	anbi2i 446	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
19	15, 18	bitri 183	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
20	14, 19	bitri 183	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
21	20	abbii 2210	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
22		rnopab 4714	. . . . 5 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
23		df-rab 2379	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
24	21, 22, 23	3eqtr4i 2125	. . . 4 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
25	10, 24	eqtri 2115	. . 3 ⊢ ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
26	8, 25	eqtri 2115	. 2 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
27	7, 26	eqtri 2115	1 ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}

Syntax hints:   ∧ wa 103   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081  {crab 2374  {copab 3920   ↦ cmpt 3921  ^◡ccnv 4466  ran crn 4468   ↾ cres 4469   “ cima 4470

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-mpt 3923  df-xp 4473  df-rel 4474  df-cnv 4475  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480

This theorem is referenced by:  mptiniseg  4959  dmmpt  4960  fmpt  5488  f1oresrab  5502  suppssfv  5890  suppssov1  5891  infrenegsupex  9181  infxrnegsupex  10806