Theorem f1od 5726

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

Hypotheses

Ref	Expression
f1od.1	⊢ F = (x ∈ A ↦ C)
f1od.2	⊢ ((φ ∧ x ∈ A) → C ∈ W)
f1od.3	⊢ ((φ ∧ y ∈ B) → D ∈ X)
f1od.4	⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D)))

Assertion

Ref	Expression
f1od	⊢ (φ → F:A–1-1-onto→B)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D   φ,x,y

Allowed substitution hints:   C(x)   D(y)   F(x,y)   W(x,y)   X(x,y)

Proof of Theorem f1od

Step	Hyp	Ref	Expression
1		f1od.2	. . . 4 ⊢ ((φ ∧ x ∈ A) → C ∈ W)
2	1	ralrimiva 2697	. . 3 ⊢ (φ → ∀x ∈ A C ∈ W)
3		f1od.1	. . . 4 ⊢ F = (x ∈ A ↦ C)
4	3	fnmpt 5689	. . 3 ⊢ (∀x ∈ A C ∈ W → F Fn A)
5	2, 4	syl 15	. 2 ⊢ (φ → F Fn A)
6		f1od.3	. . . . 5 ⊢ ((φ ∧ y ∈ B) → D ∈ X)
7	6	ralrimiva 2697	. . . 4 ⊢ (φ → ∀y ∈ B D ∈ X)
8		eqid 2353	. . . . 5 ⊢ (y ∈ B ↦ D) = (y ∈ B ↦ D)
9	8	fnmpt 5689	. . . 4 ⊢ (∀y ∈ B D ∈ X → (y ∈ B ↦ D) Fn B)
10	7, 9	syl 15	. . 3 ⊢ (φ → (y ∈ B ↦ D) Fn B)
11		f1od.4	. . . . . 6 ⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D)))
12	11	opabbidv 4625	. . . . 5 ⊢ (φ → {⟨y, x⟩ ∣ (x ∈ A ∧ y = C)} = {⟨y, x⟩ ∣ (y ∈ B ∧ x = D)})
13		df-mpt 5652	. . . . . . . 8 ⊢ (x ∈ A ↦ C) = {⟨x, y⟩ ∣ (x ∈ A ∧ y = C)}
14	3, 13	eqtri 2373	. . . . . . 7 ⊢ F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = C)}
15	14	cnveqi 4887	. . . . . 6 ⊢ ◡F = ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = C)}
16		cnvopab 5030	. . . . . 6 ⊢ ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = C)} = {⟨y, x⟩ ∣ (x ∈ A ∧ y = C)}
17	15, 16	eqtri 2373	. . . . 5 ⊢ ◡F = {⟨y, x⟩ ∣ (x ∈ A ∧ y = C)}
18		df-mpt 5652	. . . . 5 ⊢ (y ∈ B ↦ D) = {⟨y, x⟩ ∣ (y ∈ B ∧ x = D)}
19	12, 17, 18	3eqtr4g 2410	. . . 4 ⊢ (φ → ◡F = (y ∈ B ↦ D))
20	19	fneq1d 5175	. . 3 ⊢ (φ → (◡F Fn B ↔ (y ∈ B ↦ D) Fn B))
21	10, 20	mpbird 223	. 2 ⊢ (φ → ◡F Fn B)
22		dff1o4 5294	. 2 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))
23	5, 21, 22	sylanbrc 645	1 ⊢ (φ → F:A–1-1-onto→B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  {copab 4622  ^◡ccnv 4771   Fn wfn 4776  –1-1-onto→wf1o 4780   ↦ cmpt 5651

This theorem was proved from 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 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-f1 4792  df-fo 4793  df-f1o 4794  df-mpt 5652

This theorem is referenced by:  f1o2d  5727