Theorem fmpt 5693

Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
fmpt.1	⊢ F = (x ∈ A ↦ C)

Assertion

Ref	Expression
fmpt	⊢ (∀x ∈ A C ∈ B ↔ F:A–→B)

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   C(x)   F(x)

Proof of Theorem fmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpt.1	. . . 4 ⊢ F = (x ∈ A ↦ C)
2	1	fnmpt 5690	. . 3 ⊢ (∀x ∈ A C ∈ B → F Fn A)
3	1	rnmpt 5687	. . . 4 ⊢ ran F = {y ∣ ∃x ∈ A y = C}
4		r19.29 2755	. . . . . . 7 ⊢ ((∀x ∈ A C ∈ B ∧ ∃x ∈ A y = C) → ∃x ∈ A (C ∈ B ∧ y = C))
5		eleq1 2413	. . . . . . . . 9 ⊢ (y = C → (y ∈ B ↔ C ∈ B))
6	5	biimparc 473	. . . . . . . 8 ⊢ ((C ∈ B ∧ y = C) → y ∈ B)
7	6	rexlimivw 2735	. . . . . . 7 ⊢ (∃x ∈ A (C ∈ B ∧ y = C) → y ∈ B)
8	4, 7	syl 15	. . . . . 6 ⊢ ((∀x ∈ A C ∈ B ∧ ∃x ∈ A y = C) → y ∈ B)
9	8	ex 423	. . . . 5 ⊢ (∀x ∈ A C ∈ B → (∃x ∈ A y = C → y ∈ B))
10	9	abssdv 3341	. . . 4 ⊢ (∀x ∈ A C ∈ B → {y ∣ ∃x ∈ A y = C} ⊆ B)
11	3, 10	syl5eqss 3316	. . 3 ⊢ (∀x ∈ A C ∈ B → ran F ⊆ B)
12		df-f 4792	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
13	2, 11, 12	sylanbrc 645	. 2 ⊢ (∀x ∈ A C ∈ B → F:A–→B)
14	1	mptpreima 5683	. . . 4 ⊢ (◡F “ B) = {x ∈ A ∣ C ∈ B}
15		fimacnv 5412	. . . 4 ⊢ (F:A–→B → (◡F “ B) = A)
16	14, 15	syl5reqr 2400	. . 3 ⊢ (F:A–→B → A = {x ∈ A ∣ C ∈ B})
17		rabid2 2789	. . 3 ⊢ (A = {x ∈ A ∣ C ∈ B} ↔ ∀x ∈ A C ∈ B)
18	16, 17	sylib 188	. 2 ⊢ (F:A–→B → ∀x ∈ A C ∈ B)
19	13, 18	impbii 180	1 ⊢ (∀x ∈ A C ∈ B ↔ F:A–→B)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  {crab 2619   ⊆ wss 3258   “ cima 4723  ^◡ccnv 4772  ran crn 4774   Fn wfn 4777  –→wf 4778   ↦ cmpt 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-mpt 5653

This theorem is used by:  fmpti  5694  fmptd  5695  fmpt2x  5731  enprmaplem5  6081