Theorem dom3d 6190

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)

Hypotheses

Ref	Expression
dom2d.1	⊢ (φ → (x ∈ A → 𝐶 ∈ B))
dom2d.2	⊢ (φ → ((x ∈ A ∧ y ∈ A) → (𝐶 = 𝐷 ↔ x = y)))
dom3d.3	⊢ (φ → A ∈ 𝑉)
dom3d.4	⊢ (φ → B ∈ 𝑊)

Assertion

Ref	Expression
dom3d	⊢ (φ → A ≼ B)

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

Allowed substitution hints:   𝐶(x)   𝐷(y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem dom3d

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dom2d.1	. . . . . 6 ⊢ (φ → (x ∈ A → 𝐶 ∈ B))
2		dom2d.2	. . . . . 6 ⊢ (φ → ((x ∈ A ∧ y ∈ A) → (𝐶 = 𝐷 ↔ x = y)))
3	1, 2	dom2lem 6188	. . . . 5 ⊢ (φ → (x ∈ A ↦ 𝐶):A–1-1→B)
4		f1f 5035	. . . . 5 ⊢ ((x ∈ A ↦ 𝐶):A–1-1→B → (x ∈ A ↦ 𝐶):A⟶B)
5	3, 4	syl 14	. . . 4 ⊢ (φ → (x ∈ A ↦ 𝐶):A⟶B)
6		dom3d.3	. . . 4 ⊢ (φ → A ∈ 𝑉)
7		dom3d.4	. . . 4 ⊢ (φ → B ∈ 𝑊)
8		fex2 5002	. . . 4 ⊢ (((x ∈ A ↦ 𝐶):A⟶B ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → (x ∈ A ↦ 𝐶) ∈ V)
9	5, 6, 7, 8	syl3anc 1134	. . 3 ⊢ (φ → (x ∈ A ↦ 𝐶) ∈ V)
10		f1eq1 5030	. . . 4 ⊢ (z = (x ∈ A ↦ 𝐶) → (z:A–1-1→B ↔ (x ∈ A ↦ 𝐶):A–1-1→B))
11	10	spcegv 2635	. . 3 ⊢ ((x ∈ A ↦ 𝐶) ∈ V → ((x ∈ A ↦ 𝐶):A–1-1→B → ∃z z:A–1-1→B))
12	9, 3, 11	sylc 56	. 2 ⊢ (φ → ∃z z:A–1-1→B)
13		brdomg 6165	. . 3 ⊢ (B ∈ 𝑊 → (A ≼ B ↔ ∃z z:A–1-1→B))
14	7, 13	syl 14	. 2 ⊢ (φ → (A ≼ B ↔ ∃z z:A–1-1→B))
15	12, 14	mpbird 156	1 ⊢ (φ → A ≼ B)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755   ↦ cmpt 3809  ⟶wf 4841  –1-1→wf1 4842   ≼ cdom 6156

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-13 1401  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  ax-un 4136

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  df-dom 6159

This theorem is referenced by:  dom3  6192  xpdom2  6241  fopwdom  6246