Theorem dom2lem 7881

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 2948	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3		eqid 2610	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fmpt 6289	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	2, 4	sylib 207	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6	1	imp 444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7	3	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
8	7	adantll 746	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
9	6, 8	mpdan 699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
10	9	adantrr 749	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
11		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
12		nffvmpt1 6111	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
13	12	nfeq1 2764	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷
14	11, 13	nfim 1813	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
15		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	15	anbi2d 736	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
17	16	imbi1d 330	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)))
18	15	anbi1d 737	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19		anidm 674	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
20	18, 19	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴))
21	20	anbi2d 736	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
22		fveq2 6103	. . . . . . . . . . . . 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 444	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
26	25	biimparc 503	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → 𝐶 = 𝐷)
27	23, 26	eqeq12d 2625	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷))
28	27	ex 449	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
29	21, 28	sylbird 249	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
30	29	pm5.74d 261	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
31	17, 30	bitrd 267	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
32	14, 31, 9	chvar 2250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
33	32	adantrl 748	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
34	10, 33	eqeq12d 2625	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
35	25	biimpd 218	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 → 𝑥 = 𝑦))
36	34, 35	sylbid 229	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
37	36	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
38		nfmpt1 4675	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
39		nfcv 2751	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ 𝐶)
40	38, 39	dff13f 6417	. 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 695	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812

This theorem is referenced by:  dom2d  7882  dom3d  7883  ixpfi2  8147  infxpenc2lem1  8725  dfac12lem2  8849  4sqlem11  15497  odf1o1  17810  odf1o2  17811  dis2ndc  21073  hauspwpwf1  21601  itg1addlem4  23272  basellem3  24609  fsumvma  24738  dchrisum0fno1  25000