Theorem dom2lem 6738

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 2538	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3		eqid 2165	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fmpt 5635	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	2, 4	sylib 121	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6	1	imp 123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7	3	fvmpt2 5569	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
8	7	adantll 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
9	6, 8	mpdan 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
10	9	adantrr 471	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
11		nfv 1516	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
12		nffvmpt1 5497	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
13	12	nfeq1 2318	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷
14	11, 13	nfim 1560	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
15		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	15	anbi2d 460	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
17	16	imbi1d 230	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)))
18	15	anbi1d 461	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19		anidm 394	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
20	18, 19	bitrdi 195	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴))
21	20	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
22		fveq2 5486	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
23	22	adantr 274	. . . . . . . . . . . 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 123	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
26	25	biimparc 297	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → 𝐶 = 𝐷)
27	23, 26	eqeq12d 2180	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷))
28	27	ex 114	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
29	21, 28	sylbird 169	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
30	29	pm5.74d 181	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
31	17, 30	bitrd 187	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
32	14, 31, 9	chvar 1745	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
33	32	adantrl 470	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
34	10, 33	eqeq12d 2180	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
35	25	biimpd 143	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 → 𝑥 = 𝑦))
36	34, 35	sylbid 149	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
37	36	ralrimivva 2548	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
38		nfmpt1 4075	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
39		nfcv 2308	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ 𝐶)
40	38, 39	dff13f 5738	. 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 414	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444   ↦ cmpt 4043  ⟶wf 5184  –1-1→wf1 5185  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fv 5196

This theorem is referenced by:  dom2d  6739  dom3d  6740