Theorem dom2lem 6766

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 2549	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
3		eqid 2177	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
4	3	fmpt 5662	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
5	2, 4	sylib 122	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵)
6	1	imp 124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7	3	fvmpt2 5595	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
8	7	adantll 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
9	6, 8	mpdan 421	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
10	9	adantrr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
11		nfv 1528	. . . . . . . 8 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴)
12		nffvmpt1 5522	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
13	12	nfeq1 2329	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷
14	11, 13	nfim 1572	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
15		eleq1 2240	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16	15	anbi2d 464	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
17	16	imbi1d 231	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)))
18	15	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19		anidm 396	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴)
20	18, 19	bitrdi 196	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴))
21	20	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)))
22		fveq2 5511	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
23	22	adantr 276	. . . . . . . . . . . 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 124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))
26	25	biimparc 299	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → 𝐶 = 𝐷)
27	23, 26	eqeq12d 2192	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷))
28	27	ex 115	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
29	21, 28	sylbird 170	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
30	29	pm5.74d 182	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
31	17, 30	bitrd 188	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)))
32	14, 31, 9	chvar 1757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
33	32	adantrl 478	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = 𝐷)
34	10, 33	eqeq12d 2192	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
35	25	biimpd 144	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝐶 = 𝐷 → 𝑥 = 𝑦))
36	34, 35	sylbid 150	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
37	36	ralrimivva 2559	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) → 𝑥 = 𝑦))
38		nfmpt1 4093	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
39		nfcv 2319	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ 𝐶)
40	38, 39	dff13f 5765	. 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 417	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ↦ cmpt 4061  ⟶wf 5208  –1-1→wf1 5209  ‘cfv 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fv 5220

This theorem is referenced by:  dom2d  6767  dom3d  6768