Theorem dom2lem 6188

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 2385	. . 3 ⊢ (φ → ∀x ∈ A 𝐶 ∈ B)
3		eqid 2037	. . . 4 ⊢ (x ∈ A ↦ 𝐶) = (x ∈ A ↦ 𝐶)
4	3	fmpt 5262	. . 3 ⊢ (∀x ∈ A 𝐶 ∈ B ↔ (x ∈ A ↦ 𝐶):A⟶B)
5	2, 4	sylib 127	. 2 ⊢ (φ → (x ∈ A ↦ 𝐶):A⟶B)
6	1	imp 115	. . . . . . 7 ⊢ ((φ ∧ x ∈ A) → 𝐶 ∈ B)
7	3	fvmpt2 5197	. . . . . . . 8 ⊢ ((x ∈ A ∧ 𝐶 ∈ B) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶)
8	7	adantll 445	. . . . . . 7 ⊢ (((φ ∧ x ∈ A) ∧ 𝐶 ∈ B) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶)
9	6, 8	mpdan 398	. . . . . 6 ⊢ ((φ ∧ x ∈ A) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶)
10	9	adantrr 448	. . . . 5 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶)
11		nfv 1418	. . . . . . . 8 ⊢ Ⅎx(φ ∧ y ∈ A)
12		nffvmpt1 5129	. . . . . . . . 9 ⊢ Ⅎx((x ∈ A ↦ 𝐶)‘y)
13	12	nfeq1 2184	. . . . . . . 8 ⊢ Ⅎx((x ∈ A ↦ 𝐶)‘y) = 𝐷
14	11, 13	nfim 1461	. . . . . . 7 ⊢ Ⅎx((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘y) = 𝐷)
15		eleq1 2097	. . . . . . . . . 10 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
16	15	anbi2d 437	. . . . . . . . 9 ⊢ (x = y → ((φ ∧ x ∈ A) ↔ (φ ∧ y ∈ A)))
17	16	imbi1d 220	. . . . . . . 8 ⊢ (x = y → (((φ ∧ x ∈ A) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶) ↔ ((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶)))
18	15	anbi1d 438	. . . . . . . . . . . 12 ⊢ (x = y → ((x ∈ A ∧ y ∈ A) ↔ (y ∈ A ∧ y ∈ A)))
19		anidm 376	. . . . . . . . . . . 12 ⊢ ((y ∈ A ∧ y ∈ A) ↔ y ∈ A)
20	18, 19	syl6bb 185	. . . . . . . . . . 11 ⊢ (x = y → ((x ∈ A ∧ y ∈ A) ↔ y ∈ A))
21	20	anbi2d 437	. . . . . . . . . 10 ⊢ (x = y → ((φ ∧ (x ∈ A ∧ y ∈ A)) ↔ (φ ∧ y ∈ A)))
22		fveq2 5121	. . . . . . . . . . . . 13 ⊢ (x = y → ((x ∈ A ↦ 𝐶)‘x) = ((x ∈ A ↦ 𝐶)‘y))
23	22	adantr 261	. . . . . . . . . . . 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 115	. . . . . . . . . . . . 13 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (𝐶 = 𝐷 ↔ x = y))
26	25	biimparc 283	. . . . . . . . . . . 12 ⊢ ((x = y ∧ (φ ∧ (x ∈ A ∧ y ∈ A))) → 𝐶 = 𝐷)
27	23, 26	eqeq12d 2051	. . . . . . . . . . 11 ⊢ ((x = y ∧ (φ ∧ (x ∈ A ∧ y ∈ A))) → (((x ∈ A ↦ 𝐶)‘x) = 𝐶 ↔ ((x ∈ A ↦ 𝐶)‘y) = 𝐷))
28	27	ex 108	. . . . . . . . . 10 ⊢ (x = y → ((φ ∧ (x ∈ A ∧ y ∈ A)) → (((x ∈ A ↦ 𝐶)‘x) = 𝐶 ↔ ((x ∈ A ↦ 𝐶)‘y) = 𝐷)))
29	21, 28	sylbird 159	. . . . . . . . 9 ⊢ (x = y → ((φ ∧ y ∈ A) → (((x ∈ A ↦ 𝐶)‘x) = 𝐶 ↔ ((x ∈ A ↦ 𝐶)‘y) = 𝐷)))
30	29	pm5.74d 171	. . . . . . . 8 ⊢ (x = y → (((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶) ↔ ((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘y) = 𝐷)))
31	17, 30	bitrd 177	. . . . . . 7 ⊢ (x = y → (((φ ∧ x ∈ A) → ((x ∈ A ↦ 𝐶)‘x) = 𝐶) ↔ ((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘y) = 𝐷)))
32	14, 31, 9	chvar 1637	. . . . . 6 ⊢ ((φ ∧ y ∈ A) → ((x ∈ A ↦ 𝐶)‘y) = 𝐷)
33	32	adantrl 447	. . . . 5 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → ((x ∈ A ↦ 𝐶)‘y) = 𝐷)
34	10, 33	eqeq12d 2051	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (((x ∈ A ↦ 𝐶)‘x) = ((x ∈ A ↦ 𝐶)‘y) ↔ 𝐶 = 𝐷))
35	25	biimpd 132	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (𝐶 = 𝐷 → x = y))
36	34, 35	sylbid 139	. . 3 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ A)) → (((x ∈ A ↦ 𝐶)‘x) = ((x ∈ A ↦ 𝐶)‘y) → x = y))
37	36	ralrimivva 2395	. 2 ⊢ (φ → ∀x ∈ A ∀y ∈ A (((x ∈ A ↦ 𝐶)‘x) = ((x ∈ A ↦ 𝐶)‘y) → x = y))
38		nfmpt1 3841	. . 3 ⊢ Ⅎx(x ∈ A ↦ 𝐶)
39		nfcv 2175	. . 3 ⊢ Ⅎy(x ∈ A ↦ 𝐶)
40	38, 39	dff13f 5352	. 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 394	1 ⊢ (φ → (x ∈ A ↦ 𝐶):A–1-1→B)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ↦ cmpt 3809  ⟶wf 4841  –1-1→wf1 4842  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853

This theorem is referenced by:  dom2d  6189  dom3d  6190