Theorem fmpt 5692

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 5689	. . 3 ⊢ (∀x ∈ A C ∈ B → F Fn A)
3	1	rnmpt 5686	. . . 4 ⊢ ran F = {y ∣ ∃x ∈ A y = C}
4		r19.29 2754	. . . . . . 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 2734	. . . . . . 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 3340	. . . 4 ⊢ (∀x ∈ A C ∈ B → {y ∣ ∃x ∈ A y = C} ⊆ B)
11	3, 10	syl5eqss 3315	. . 3 ⊢ (∀x ∈ A C ∈ B → ran F ⊆ B)
12		df-f 4791	. . 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 5682	. . . 4 ⊢ (◡F “ B) = {x ∈ A ∣ C ∈ B}
15		fimacnv 5411	. . . 4 ⊢ (F:A–→B → (◡F “ B) = A)
16	14, 15	syl5reqr 2400	. . 3 ⊢ (F:A–→B → A = {x ∈ A ∣ C ∈ B})
17		rabid2 2788	. . 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)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  {crab 2618   ⊆ wss 3257   “ cima 4722  ^◡ccnv 4771  ran crn 4773   Fn wfn 4776  –→wf 4777   ↦ cmpt 5651

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-mpt 5652

This theorem is referenced by:  fmpti  5693  fmptd  5694  fmpt2x  5730  enprmaplem5  6080