Theorem dom2lem 9052

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)

Hypotheses

Ref	Expression
dom2d.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
dom2d.2	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))

Assertion

Ref	Expression
dom2lem	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)

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

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem dom2lem

Step	Hyp	Ref	Expression
1		dom2d.1	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))
2	1	ralrimiv 3151	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fmpt 7144	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	2, 4	sylib 218	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6	1	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7	3	fvmpt2 7040	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
8	7	adantll 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
9	6, 8	mpdan 686	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
10	9	adantrr 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
11		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
12		nffvmpt1 6931	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
13	12	nfeq1 2924	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷
14	11, 13	nfim 1895	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
15		eleq1w 2827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	15	anbi2d 629	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
17	16	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)))
18	15	anbi1d 630	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19		anidm 564	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
20	18, 19	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴))
21	20	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
22		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
24		dom2d.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)))
25	24	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
26	25	biimparc 479	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → 𝐶 = 𝐷)
27	23, 26	eqeq12d 2756	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷))
28	27	ex 412	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
29	21, 28	sylbird 260	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
30	29	pm5.74d 273	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
31	17, 30	bitrd 279	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
32	14, 31, 9	chvarfv 2241	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
33	32	adantrl 715	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
34	10, 33	eqeq12d 2756	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
35	25	biimpd 229	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 → 𝑥 = 𝑦))
36	34, 35	sylbid 240	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
37	36	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
38		nfmpt1 5274	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
39		nfcv 2908	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ 𝐶)
40	38, 39	dff13f 7293	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦)))
41	5, 37, 40	sylanbrc 582	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fv 6581

This theorem is referenced by:  dom2d  9053  dom3d  9054  ixpfi2  9420  infxpenc2lem1  10088  dfac12lem2  10214  4sqlem11  17002  odf1o1  19614  odf1o2  19615  dis2ndc  23489  hauspwpwf1  24016  itg1addlem4  25753  itg1addlem4OLD  25754  basellem3  27144  fsumvma  27275  dchrisum0fno1  27573