Theorem dff4 5421

Description: Alternate definition of a mapping. (Contributed by set.mm contributors, 20-Mar-2007.)

Assertion

Ref	Expression
dff4	⊢ (F:A–→B ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B xFy))

Distinct variable groups:   x,y,A   x,B,y   x,F,y

Proof of Theorem dff4

Step	Hyp	Ref	Expression
1		dff3 5420	. 2 ⊢ (F:A–→B ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy))
2		brelrn 4960	. . . . . . . 8 ⊢ (xFy → y ∈ ran F)
3		rnss 4959	. . . . . . . . . 10 ⊢ (F ⊆ (A × B) → ran F ⊆ ran (A × B))
4		rnxpss 5053	. . . . . . . . . 10 ⊢ ran (A × B) ⊆ B
5	3, 4	syl6ss 3284	. . . . . . . . 9 ⊢ (F ⊆ (A × B) → ran F ⊆ B)
6	5	sseld 3272	. . . . . . . 8 ⊢ (F ⊆ (A × B) → (y ∈ ran F → y ∈ B))
7	2, 6	syl5 28	. . . . . . 7 ⊢ (F ⊆ (A × B) → (xFy → y ∈ B))
8	7	pm4.71rd 616	. . . . . 6 ⊢ (F ⊆ (A × B) → (xFy ↔ (y ∈ B ∧ xFy)))
9	8	eubidv 2212	. . . . 5 ⊢ (F ⊆ (A × B) → (∃!y xFy ↔ ∃!y(y ∈ B ∧ xFy)))
10		df-reu 2621	. . . . 5 ⊢ (∃!y ∈ B xFy ↔ ∃!y(y ∈ B ∧ xFy))
11	9, 10	syl6bbr 254	. . . 4 ⊢ (F ⊆ (A × B) → (∃!y xFy ↔ ∃!y ∈ B xFy))
12	11	ralbidv 2634	. . 3 ⊢ (F ⊆ (A × B) → (∀x ∈ A ∃!y xFy ↔ ∀x ∈ A ∃!y ∈ B xFy))
13	12	pm5.32i 618	. 2 ⊢ ((F ⊆ (A × B) ∧ ∀x ∈ A ∃!y xFy) ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B xFy))
14	1, 13	bitri 240	1 ⊢ (F:A–→B ↔ (F ⊆ (A × B) ∧ ∀x ∈ A ∃!y ∈ B xFy))

Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∃!weu 2204  ∀wral 2614  ∃!wreu 2616   ⊆ wss 3257   class class class wbr 4639   × cxp 4770  ran crn 4773  –→wf 4777

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-fun 4789  df-fn 4790  df-f 4791

This theorem is referenced by: (None)