Theorem fmpt2d 5695

Description: Domain and co-domain of the mapping operation; deduction form. (Contributed by set.mm contributors, 9-Apr-2013.)

Hypotheses

Ref	Expression
fmpt2d.1	⊢ (φ → (x ∈ A → B ∈ V))
fmpt2d.2	⊢ F = (x ∈ A ↦ B)
fmpt2d.3	⊢ (φ → (y ∈ A → (F ‘y) ∈ C))

Assertion

Ref	Expression
fmpt2d	⊢ (φ → F:A–→C)

Distinct variable groups:   x,y,A   y,C   y,F   φ,x,y

Allowed substitution hints:   B(x,y)   C(x)   F(x)   V(x,y)

Proof of Theorem fmpt2d

Step	Hyp	Ref	Expression
1		fmpt2d.1	. . . 4 ⊢ (φ → (x ∈ A → B ∈ V))
2	1	ralrimiv 2696	. . 3 ⊢ (φ → ∀x ∈ A B ∈ V)
3		fmpt2d.2	. . . 4 ⊢ F = (x ∈ A ↦ B)
4	3	fnmpt 5689	. . 3 ⊢ (∀x ∈ A B ∈ V → F Fn A)
5	2, 4	syl 15	. 2 ⊢ (φ → F Fn A)
6		fmpt2d.3	. . . 4 ⊢ (φ → (y ∈ A → (F ‘y) ∈ C))
7	6	ralrimiv 2696	. . 3 ⊢ (φ → ∀y ∈ A (F ‘y) ∈ C)
8		fnfvrnss 5429	. . 3 ⊢ ((F Fn A ∧ ∀y ∈ A (F ‘y) ∈ C) → ran F ⊆ C)
9	5, 7, 8	syl2anc 642	. 2 ⊢ (φ → ran F ⊆ C)
10		df-f 4791	. 2 ⊢ (F:A–→C ↔ (F Fn A ∧ ran F ⊆ C))
11	5, 9, 10	sylanbrc 645	1 ⊢ (φ → F:A–→C)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777   ‘cfv 4781   ↦ 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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-mpt 5652

This theorem is referenced by: (None)