Theorem f1o3d 31851

Description: Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)

Hypotheses

Ref	Expression
f1o3d.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶))
f1o3d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
f1o3d.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
f1o3d.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))

Assertion

Ref	Expression
f1o3d	⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

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)

Proof of Theorem f1o3d

Step	Hyp	Ref	Expression
1		f1o3d.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
2	1	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3		eqid 2733	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fnmpt 6691	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
6		f1o3d.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶))
7	6	fneq1d 6643	. . . 4 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
8	5, 7	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9		f1o3d.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
10	9	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝐴)
11		eqid 2733	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷)
12	11	fnmpt 6691	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐷 ∈ 𝐴 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)
13	10, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)
14		eleq1a 2829	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
15	1, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵))
16	15	impr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵)
17		f1o3d.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))
18	17	biimpar 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
19	18	exp42 437	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑥 = 𝐷))))
20	19	com34 91	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷))))
21	20	imp32 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷))
22	16, 21	jcai 518	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))
23		eleq1a 2829	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴))
24	9, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴))
25	24	impr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴)
26	17	biimpa 478	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
27	26	exp42 437	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → 𝑦 = 𝐶))))
28	27	com23 86	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑦 = 𝐶))))
29	28	com34 91	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶))))
30	29	imp32 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶))
31	25, 30	jcai 518	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))
32	22, 31	impbida 800	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))
33	32	opabbidv 5215	. . . . . 6 ⊢ (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)})
34		df-mpt 5233	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
35	6, 34	eqtrdi 2789	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
36	35	cnveqd 5876	. . . . . . 7 ⊢ (𝜑 → ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
37		cnvopab 6139	. . . . . . 7 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
38	36, 37	eqtrdi 2789	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
39		df-mpt 5233	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}
40	39	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)})
41	33, 38, 40	3eqtr4d 2783	. . . . 5 ⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))
42	41	fneq1d 6643	. . . 4 ⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵))
43	13, 42	mpbird 257	. . 3 ⊢ (𝜑 → ◡𝐹 Fn 𝐵)
44		dff1o4 6842	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
45	8, 43, 44	sylanbrc 584	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
46	45, 41	jca 513	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {copab 5211   ↦ cmpt 5232  ^◡ccnv 5676   Fn wfn 6539  –1-1-onto→wf1o 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551

This theorem is referenced by:  fmptco1f1o  31857  ballotlemsf1o  33512